I am not a professional photographer, I am just taking pictures as a hobby. I take portraits, landscapes, and some wedding pictures. I am using a Nikon D750 camera, should I be shooting in jpeg fine or raw?
Ultimately, which format is a better fit for you is up to you, and what/how you shoot and process your images.
Some folks will categorize the difference as amateur (JPEG) and pro (RAW), or not-serious (JPEG) and serious (RAW), but it's simply a matter of which format gets you what you want out of an image.
Why not both?
Side note: space is cheap these days; cards and hard drives are among the less-expensive hardware bits a photographer can buy. I'd actually recommend that you try shooting in RAW+JPEG, so you have the benefits of both formats available to you, or at least so you can try both and learn which one is more to your taste with identical images to side-by-side. It won't affect much, unless you're shooting superfast action and can't burst fast enough in JPEG+RAW. Most cameras offer you the option of saving the RAW file as it makes the JPEG.
JPEGs are smaller compressed files and an internet standard, so they require no processing to achieve a deliverable file. And most in-camera processing features only affect a JPEG file. So, if you need to deliver a lot of photos quickly (say, as a professional who needs to deliver a lot of images quickly, or a single shot before the competition can). Or, say if you're an instant-gratification social media creature, you may prefer the immediacy of a JPEG file straight from the camera.
But because JPEG is a lossy format, some of the color data is discarded in the compression scheme. At the "Fine" setting, you're retaining the most your camera allows. But nailing the exposure and any in-camera processing choices become more important. You can still post-process the image. You just won't have the same latitude on edits that RAW would give you, particularly for some color shifts (e.g., white balance), because the lost data JPEG dumped in the compression can leave behind artifacts with certain kinds of adjustments. You have to have mastered your camera and its settings so that you know you're getting what you want in the camera, or at least close enough that you can post-process without creating artifacts.
That doesn't mean you can only do mild processing. Some in-camera processing can get pretty wild (think: tilt-shift, low/high-key effects, and everything else a scene mode can accomplish in-camera). JPEG engines are where the camera manufacturers spent time and money developing proprietary algorithms, and sometimes the look they can achieve may be difficult to duplicate on your own (go google "Fuji Classic Chrome" ;-), and could be a look you really want. Back in the early days of dSLRs, in-camera processing was bad enough that even as a beginning amateur you could process the images better yourself. Today, this may no longer be the case. So if you cherish how the images look on the LCD, a JPEG might be the only way to get it, and will certainly be the easiest way to get it.
And, obviously, if you prefer not to post-process at all, then JPEG is your friend.
RAW files are (mostly) raw data dumps from the sensor. Obviously, the data goes through the camera processor, and there may actually be some resizing (sRAW), compression or processing going on, but for the most part, the file is uncooked. However, RAW is not a specific file format or file standard. It's not even an acronym. RAW file formats change with each individual camera model. The .NEF files coming out of your D750 are not in an identical format to the .NEF files coming out of a D3500 or a D700. So, you need software that can grok your D750's RAW files to do RAW conversion. JPEG editors typically don't need to be kept up-to-date, and you may have to wait for an update in your converter to use a RAW file directly, if you're buying a just-released camera model. (See DNG converter doesn't convert, Photoshop CS2 doesn't open DNG for information on the "Photoshop Tax").
RAW files contain all the data the sensor gathered in a gloriously lossless format. So, while the files are larger than JPEGs, they also allow you to do far more dramatic changes to the color data without showing artifacts as quickly as a JPEG file will. So if you like to go hog wild with sliders in post or do more extreme forms of processing, such as HDR, cross-processing, or compositing, or there's a specific effect you want that your camera's processing doesn't do, RAW is typically going to be a more flexible format to play with, if you have the time and willingness to do so. So, RAW is often seen as the "serious" or "professional" format to shoot in.
But RAW can also be better for "fixing" shooting mistakes--particularly white balancing and exposure issues. So, some might actually characterize RAW as the better, more forgiving, format for beginners who may not be able to nail exposure and white balance in-camera the way they want. RAW can act as your "safety net", as well as giving you more toys to play with when it comes to post-processing.
If the pictures that come out of your camera look good to you and you don't want to put in the effort possible with RAW images to process them yourself and make them better than your camera can give you -- by all means, use JPEG.
If you're happy with the results that's what matters. Don't worry about any of the geeky stuff, if you're getting the images you want, that's the important thing.
That said, if down the road you decide to get more serious about photography and you start wanting to get more involved in controlling the look of your images, then you need to switch to RAW and learn the post-processing piece then you need to be aware taht the images you take in JPEG will have limited ability to be worked on. When I made the switch from JPEG to RAW around 2005, I ended up with about four years of images I really can't use because they were in JPEG and the work needed to bring them up to snuff wasn't possible.
Also realize those JEPGS are fine for printing and sharing online, but probably not for any "serious" use where you might want to sell them. Since that's not part of your interest, don't worry about it but I wanted to point it out.
As long as you're aware of that possible tradeoffs down the road, do what lets you enjoy taking pictures. It's perfectly okay to change your mind later if you get more serious about your photography, none of this is permanent.
If you like what you are getting, keep doing it. Only change it if you feel you ought to do better and want to. Don't let the advice of what some like to try to tell photographers is the "way things ought to be" get in the way. Do what makes you enjoy doing it. Don't try to do more than that until you feel you need to because you want to get better at your hobby.
"Professional" has nothing to do with it. It comes down to whether you are willing to accept what the camera produces, or if you instead wish to have some degree of freedom to adjust the camera's product before delivering the final results.
JPEG is carefully designed to be "good enough" for almost all presentation purposes.¹ But that's the thing: it is designed only for the case of delivering a finished photograph to the one who wishes to view it. When you ask a raw-capable camera to deliver JPEGs instead, you are telling it that you will accept whatever it produces.
Obviously JPEGs aren't entirely un-editable, but there are costs to almost every kind of edit you can name:
Lighting adjustments: This is the biggie. JPEG has only 8 bits of brightness information. (That is, 2⁸, or 256 levels of gray.) There is nothing below visible black in the image, and nothing above visible white in the image. Thus, if it is overexposed or underexposed, any brightness adjustments will begin to produce banding and other bad artifacts.
Most any raw-capable camera will have at least 2¹⁰ levels of gray per color channel, and often more. This gets you thousands or tens of thousands of levels of gray to play with. When you're pushing the lighting levels around, the more levels you have to play with, the less risk you have of producing something unusable.
Color temperature adjustments: The auto-white balance feature of modern cameras is amazing in its ability to guess the correct white balance, but it isn't perfect. Since after-capture white balance adjustments require the same sorts of available dynamic range as lighting adjustments, if there is too much error in the camera's white balance setting, you will not have the dynamic range available to fix it.
Cropping, rotation, etc.: It's better to do such adjustments to a raw image than a JPEG because it avoids a recompression step. Each recompression step loses more information, except in very uncommon cases.
It therefore comes down to a question of time: will you take the time to import all your raw images into a photo management application and make any necessary adjustments to them before compressing them to their final JPEG form, or do you wish to deal with only a card full of finalized JPEGs?
I do not mean my "good enough" comment to be dismissive of the potential quality of JPEG. Consider the implications of "almost all presentation purposes." I am saying that if you have an excellent source photo and render it as a JPEG, it will appear no better on most displays if you had saved it as an OpenEXR file instead, which has none of the weaknesses of JPEG. This is quite a high bar.
My point is simply that if you start with a less-than-perfect JPEG, your ability to fix it is hampered by the nature of JPEG.
Since "non-professional" is a rather broad term, it's eventually up to you to decide. But it helps making an informed decision. You'll need to understand the difference between the two formats.
JPEG is a lossy image format. It is optimized for storing photos with minimum visible distortion. However, it is lossy. Hence, if you want to change anything in your image after you've taken it, you are limited to what was preserved during the compression. Your camera may take a wrong decision, and you will loose information.
RAW is a lossless format. It saves everything seen by the matrix of your camera. Thus you can get the most out of your camera. Your computer is probably more powerful than the processor of your camera. And your brain is more capable than the automatic algorithms built into the camera. You can also spend much more time per shot than a camera can afford.
The question however is if you will want to spend this time. A good camera will usually perform very reasonable post-processing immediately, creating nice pictures.
Summary: If you don't want to bother, shoot in JPEG and save space and time. If you do want to play with and try to improve your images, shoot in RAW.
While professional photographers will use RAW pictures, for them the whole work flow from shooting pictures, processing them and getting to the final product must earn them enough money to justify the effort. If you take pictures for a hobby then you are not working under that constraint, so you have much more freedom to spend your time working with raw pictures, doing all sorts of processing on them to get to a nice result.
Both have pluses and minuses. Others already have addressed many of them, so I'll just add a few more considerations that haven't come up:
Disk space should still be a consideration. Sure, hard disks are cheap, but size still matters in many respects. Larger files means more time to back them up. More time to upload them to Dropbox or the like, if you use them. Higher bills from Dropbox etc. It can also mean the difference between fitting all wedding pictures onto a CD or DVD you burn for the couple, vs. having to split it between several.
Longevity. In the past, the limiting factor was the chemistry of the materials used. Today, it is software support. If the manufacturer discontinues a RAW format, you may end up not being able to access those files any more. Similarly, if you stored your files on hardware that is no longer supported, you may have problems. This type of digital rot is already becoming a problem. If you have Visicalc spreadsheets or an old ZIP drive (or even just a 5 1/4" floppy disk), you'll have a problem today.
Not saying that you should necessarily choose one format over the other, just a few more considerations to look at when deciding.
JPEG will be a better option for you, if and only if you take pictures to store as memories or as a hobby. Managing, storing and editing JPEG images is easier.
You will face the following difficulties with RAW images.
RAW images are much larger, it will consume a large portion of your memory card.
It is not accessible in many image viewers, shoot in RAW only if you want to edit later and make an amazing photograph.
A RAW image contains a lot of information about the photo and you can edit the photo effectively without damaging its contents or adding grain or exposure.
But again if its not for professional purposes, there's no point in putting so much effort.
RAW stores more information than JPEG and disk space is cheap, so why would you ever want to delete potentially valuable information right after taking the picture ?
If you have a machine that can create RAW then take advantage of it - otherwise it's like buying an expensive 4k video camera and setting it in 640x480 mode (remember those cheap webcams ?).
You can always get rid of that "extra information" (by converting it to JPEG) but you can't recreate it if all you have is a JPEG, which will be problematic if you happen need the RAW file later for whatever reason (editing, etc).
Check out this question for some examples of why RAW is important (or will become important, if you're still a beginner and don't have experience with RAW development software).