Learn and use both.
But before I go out and buy all the gear with no idea what I'm doing, I would just like to start off learning the correct software.
The size of your camera gear cabinet has nothing to do with this question. You do not require a $10,000 DSLR rig to justify the use of Lightroom or Photoshop.
You can use Lightroom to manage, edit, and publish the photos you already take with whatever camera you already have, even if that is only a so-so mobile phone camera. I even use Lightroom regularly on low-megapixel over-compressed JPEGs downloaded from the web.
The same goes for Photoshop.
Lightroom and Photoshop both have big learning curves
Sure, but one of those learning curves is a whole lot steeper and higher than the other.
You can learn enough to be competent in Lightroom in a few hours at most. I'm not talking about complete mastery, just basic competence. Mastery might take you years to acquire, depending on how avidly you pursue that study, but it's easy to pick up each new Lightroom element you need to know about at the time you need to know it. That is not to say that there are no hidden secrets in Lightroom, but for the most part, Lightroom's behavior is up-front and presented on a small number of separate screens, rather than hidden behind a Byzantine menu and command system in dozens of dialog boxes.
Photoshop is a whole different level of difficult. Basic competence might take a year or so to acquire, and you may never achieve true mastery.¹ I recommend watching The Photoshop Guys faithfully for at least a year, supplemented by regular doses of printed material. (Books, blogs, magazines...) Like learning a new language, total immersion learning is the fastest path to complete Photoshop mastery.
and I don't want to waste any of my time learning one when maybe the other would of been a better choice.
As with so much else, that's a false choice. It isn't bicycles or automobiles, AM radio or podcasts, fountain pens or pencils, wrenches or screwdrivers, Lightroom or Photoshop. It's "and" in all cases, each tool selected for particular purposes where it has an advantage over its alternative(s).
As a web / graphic designer...
"...photo editing is only a small part of my job."
Sorry to put words in your mouth, but I'm pretty sure they're correct ones.
To the extent that graphical manipulation is part of your job, anything not specifically about photo editing will probably have to be done in Photoshop rather than Lightroom, simply because Lightroom is not a general-purpose graphics package; Lightroom is laser-focused on photo editing.
That focus often makes Lightroom faster and/or easier to use for tasks within its design scope. A good rule of thumb is that if you have a photo that needs work, you can probably do between 95 and 100% of the work in Lightroom at least as quickly as in Photoshop, if not faster.
Anything left over — being things that either cannot reasonably be done at all in Lightroom or which can be done but only with much more difficulty than in Photoshop — can simply be left to Photoshop, via Ctrl/Cmd-E after you're done working with the photo in Lightroom. Lightroom edits are preserved in Photoshop, and vice versa.
Another way to blend the two programs is via Adobe Camera Raw, a feature of Photoshop that contains most of the functionality of Lightroom's Develop module.² That is, while in Photoshop if you need to do something that would be easy in Lightroom's Develop module, you can pop into ACR via Ctrl/Cmd-Shift-A, do your edit, and return to normal Photoshop work.³ This is usually easier than the alternatives:
Save the photo, return to Lightroom, do the Develop change there, then re-open the photo in Photoshop to continue your work.
Figure out how to the use older, more primitive⁴ Photoshop tools to achieve the same ends that Lightroom and ACR make easy.
For example, I find that making a color balance adjustment is easier via ACR than via the more idiomatic alternatives in Photoshop. That may be purely because I spend more time in Lightroom than in Photoshop these days, but then, if you follow my advice here, you will end up with the same sort of time split, so the observation should apply to you, too.
Because Lightroom can do 95-100% of your photo work without involving Photoshop, I recommend that you start learning it first, then every time you run into a wall, pop into Photoshop and learn whatever little bit you need to know over there to get past the problem.
Now, having pointed out the steep and high Photoshop learning curve, I also need to point out that it is generally much more powerful than Lightroom. For pure photo editing, there is nothing you can do in Lightroom that can't be done in Photoshop, even ignoring ACR, but the reverse is most definitely is not the case.⁵ For example, both programs have cloning and healing brushes, but Lightroom's implementation is not as powerful:
Lightroom's implementation is purely nondestructive, which means that Lightroom must play back your prior brush strokes every time you add another. By the time you get to dozens or hundreds of brush strokes, Lightroom can get really slow.
By comparison, Photoshop's implementation is destructive by default, meaning that every change is baked into the layer you make the change on. If you need something more like Lightroom's nondestructive edits, you can do your changes on a higher layer, which Photoshop can composite onto the lower layer faster than Lightroom can replay its edit history.⁶
Lightroom's nondestructive cloning and healing tool creates an ambiguity in the meaning of a click on the photo: clicks can both create new edits and select prior edits. This means you have to jump through hoops to re-edit the same portion of the image that you edited previously. Either you need to start the brush stroke slightly outside the prior edit area or you need to temporarily turn off the brush stroke outlines and pins via H so that they're temporarily unclickable.
The re-editing feature also means each edit is marked with a pin, which obscures part of the image. You either end up toggling them on and off via that same shortcut key or bouncing between the Loupe and Develop modules to get a sense of your progress toward your goals.
These are only minor hassles, but they eat away at the usage speed advantage Lightroom gains from its streamlined workflow. Compare Photoshop where prior edits are baked into pixels instantly, so you can paint over the same area as many times as you need to in order to achieve the effect you want.
A good way to see the practical differences in the two implementations is to try and clone/heal a power line out of a photo. Both programs can do this, and in fact in simple cases (e.g. power line against a clear blue sky) there is no particular advantage to using Photoshop. But try to do it in Lightroom when the power line crosses an area of high irregular detail in the photo, such as a hill covered with pine trees, and now the ability to easily make hundreds of brush strokes in a tiny area — as may be necessary in such a situation — gives Photoshop a huge advantage. As a rule, Photoshop is better at fine-detail retouching, whereas Lightroom is faster for simple, gross edits.
Lightroom's more streamlined workflow and nondestructive edit history are considerable advantages. More than once I've gone back into a photo in Lightroom's Develop module and reworked an edit, either because my skills have improved, my tastes have changed, or the tools have improved since I made the prior edit. Doing so requires no advance planning with Lightroom, whereas it does require a particular working style to pull that off in Photoshop, and that working style adds still more overhead to the process.⁷
That's just for one tool. An analysis of the pros and cons could be done anywhere both Lightroom and Photoshop have a very similar tool, and in many cases it would turn up a similarly mixed set of advantages. Thus, comparing the two programs simply on a feature catalog or bullet point list will miss these important differences.
There are also areas where the two programs don't overlap at all. I've listed most of Lightroom's advantages over Photoshop elsewhere in this answer, so I'll just list some of the areas where Photoshop has a feature that has no equivalent in Lightroom:
Lightroom lacks an equivalent for most of the contents of Photoshop's Filters menu, and in areas where there is overlap (e.g. the blur filters) Lightroom's equivalent is generally less featureful.
Lightroom can't composite images without help from a third-party plugin.
This shows up in other areas, too, such as a lack of anything like Photoshop's Calculations feature or Blend modes.
Lightroom doesn't expose the color channels in an image; it is fundamentally designed to hide or abstract such details, so that the closest you get to channels are the HSL/Saturation panels. Sometimes you've just got to have CMYK or Lab editing.
Whole classes of inherently destructive edits are simply disallowed in Lightroom, such as resizing an image with an unlocked aspect ratio. That is, Photoshop will let you take a 16:9 image and resize it to 4:3 without changing the number of vertical lines, changing the image's width only. Lightroom is aware of aspect ratios, but largely only from a print cropping standpoint, so that you can tell it how to crop a 4:3 image so it prints on 4×6" paper, which is a 2:3 aspect ratio.
Lightroom is entirely missing support for 3D graphics. Photoshop isn't an especially great 3D modeling program, but it's a pretty good texture painting program. As well, it's got a pretty good rendering engine built in, and it's great for compositing existing 3D imagery into a 2D comp without all the messing around required to merge such images in from outside 3D rendering systems.⁸
Lightroom likewise lacks the Measurements, Analysis, and Vanishing Point features, which let you manipulate and measure an image in real-world (i.e. 3D) coordinate space. Both programs will happily tell you that the image of the building you just shot will be 8" high when printed at 240dpi, but of the two, only Photoshop can tell you how many meters high the building is given the measured length of its baseline after correcting for the distortions inherent in photographic image capture.
Lightroom has adjustment brushes, but no painting brushes. Lightroom will let you recolor and otherwise manipulate the existing tonal values in the image, but Lightroom cannot create wholly new tonal values, only derivatives of what already exists in the image. The primary limit on what you can paint in Photoshop is your skill, not the tool. (Observe mastery.)
Lightroom has very little in the way of typography tools. If you want to put text on your images, you're pretty much limited to export watermarks. Photoshop's typography tools, by contrast, are weak only by comparison to those of page layout and vector graphics software, such as InDesign and Illustrator. Someone sufficiently motivated could lay out a book using only Photoshop.
Footnotes and Digressions:
This is why so many people believe Gimp is competitive with Photoshop: because they have no idea of the full extent of what Photoshop can actually do in the hands of a master, they form their opinion based on their limited skills and needs. That is, for many people both Photoshop and Gimp solve their problems adequately, so for their limited purposes, these two programs are equivalent.
I'm not saying that Gimp partisans are expressing an incorrect choice. For their subjective needs, they are making a rational choice. I'm just saying that Gimp is actually closer in terms of feature scope and power to Photoshop Elements or Pixelmator than to Photoshop CC.
In fact, ACR and Lightroom's Develop module are largely based on the same code, so that whenever you see an update to one, you can bet on an update to the other coming out soon after.
Unfortunately, most of the shortcut keys in ACR are different from those in Lightroom's Develop module, so that learning one is only of some help in learning the other. The concepts transfer, but the muscle memory doesn't.
I don't mean "primitive" in a disparaging way, but rather in the mathematical sense. That is to say, you might need to combine several tools in Photoshop to accomplish something that Lightroom offers in a single tool. The advantage in Photoshop is that you can later recombine those primitives in other ways which have no equivalent in Lightroom.
Lightroom likewise has many features not present in Photoshop, primarily ones involving metadata and publishing rather than direct pixel editing: the Map, Book, Slideshow, and Web modules, the Publish feature, hierarchical keyword tagging, batch metadata editing, etc.
This is a classic speed vs. space tradeoff, where Lightroom's Develop edit history takes less RAM than a Photoshop layer, but compositing layers is faster than replaying an edit history. Since each layer takes up a tiny fraction of the gigs of RAM in today's computers, the cost is nearly negligible these days, so on balance, it's generally worth paying that cost when you get into complicated retouching work, where Lightroom's nondestructive workflow would cause Lightroom to bog down.
For example, adding layers in Photoshop to achieve a nondestructive workflow only works well when you confine unrelated edits to separate layers, which you have to remember to create manually. Then, unless you go out of your way to rename each layer to describe the edit, you end up paying for that laziness later as you spend time trying to figure out what each layer does in a complicated multilayered Photoshop file.
Graphic artists in 2016 should no longer be required to know what a premultiplied alpha channel is, so that they may correctly choose whether they want one in a given context. Leaving such things up to Photoshop is a perfectly rational choice.