I'm very new to photography. Have a Canon T3, with the standard 18-55 lens and a 75-300 mm lens. I'm looking to do a lot of family photography (wife with 2 kids less than 2 years old). Most of the pictures will be indoors, but some will be outside during the day. I was looking at some prime lenses on Amazon, but I'm not sure what would be the best choice - some advice would be appreciated. I'm looking at a budget of $100-$150 for the lens.
Pretty sure you only have one choice for that price. Luckily, it's a good and reputable one for portraits and general low-light usage. They call it the nifty fifty: Canon EF 50mm F/1.8 II.
The bright F/1.8 aperture lets it shoot in low light and also gives a shallow depth-of-field which is often used in portraits. Plus, the 50mm focal-length, equivalent to 75mm on a full-frame, gives a flattering perspective for portraits.
With your budget you have two choices: the EF 50mm f/1.8 II or the EF 40mm f/2.8 STM.
(2019/01/20 - There is now also the EF 50mm f/1.8 STM that is as affordable as the EF 50mm f/1.8 II was, while having most of the usability of the more expensive EF 50mm f/1.4. For more, please see the comparisons here)
The 50mm has an aperture about 1 1/2 stops faster than the 40mm. This would allow you to use a faster shutter speed at the same ISO in low light. The downside to the wide aperture is that the depth of field is very shallow. If you are in close you'll need to stop down to around f/2.8 or more to get both eyes of your subject in focus in a 3/4 angle head shot. The other issue with the f/1.8 wide open is that it may be difficult to keep your very active children still enough to carefully focus as you need to do with a very wide aperture. And the focus, while accurate, is not extremely fast on this lens.
The 40mm "pancake" packs a lot into a very small, lightweight package. The biggest differences compared to the 50mm f/1.8 II are the metal mount and the STM focus system. The 50mm has a plastic mount. The 40mm is a little wide for tight head shots, but does nicely for other portraits. Even though the field of view on your T3 yields the equivalent of a 65mm lens, the perspective is the same as it would be on a full frame camera. Get too close and body parts like the nose will start to be overemphasized. This will also be true to a lesser extent with the 50mm.
For your situation with a young family my thought is the 40mm will be a better focal length for portraits that include multiple family members. The extra aperture may or may not be usable with the wider aperture of the 50mm. If it were me with a young family I would go for the EF 40mm f/2.8 STM. If you later decide to upgrade to a higher Rebel, the STM will focus much better with the T4i while shooting video. As far as a 50mm lens goes, I would try to save and buy the EF 50mm f/1.4 later on. It is a much better lens, in my opinion, than the EF 50mm f/1.8 II. Much better build quality, USM focus motor, a truly usable manual focus ring, and better bokeh from the 8 blade aperture (compared to the harsh bokeh from the 5 bladed f/1.8 at any aperture other than wide open) are some of the reasons why I recommend the EF 50mm f/1.4 over the f/1.8. I own both of them. While I carried the f/1.8 version around in my bag for many years, I rarely used it. Since upgrading to the f/1.4 I have found I pull it out more often due to the increased usability of the faster focus and the usable manual focus ring.
Having said all of that, if I were looking for an economical 50mm prime today I would carefully consider the newer EF 50mm f/1.8 STM. It is just as good optically as the f/1.4 and f/1.8 II. A lot of the build quality and usability problems of the f/1.8 II have been well addressed by the design of the F/1.8 STM. It sells for a fraction of the price of the f/1.4. The one thing that might be a downside for some users is the way the manual focusing ring works with all STM lenses. The focusing of the lens, even when manually focusing, is strictly focus-by-wire. Turning the focus ring on the barrel of the lens does nothing mechanically to move the focus elements of the lens. Instead, an electrical signal is sent to the camera body which in turn sends a command to the focus motor in the lens to move. This limits the smallest increment of focus adjustment to one "step" of the stepping motor in the lens. It also eliminates the ability to focus the lens when using "dumb" extension tubes or freelensing. You can't move the focus elements to park them when the lens is not attached to a powered up EOS camera body either.
I will recommend a different lens than the 40mm 2.8 and 50mm 1.8 in the other answers. As you wanted to take photos indoors a 50mm or 40mm lens will requier a bit of working distance on a crop sensor camera. I would suggest that you look at the ef-s 24mm 2.8. This will not requier as much space to back up and still has a resonably bright aparture of 2.8, if you are struggling with light the 50mm 1.8 might be better but in that case it might make more sence to save a bit more and get a Sigma 30mm 1.4, this is $400 new.
Regardless of which lens you buy consider buying used, I saw the sigma 30mm 1.4 for around $200 on ebay and the other lenses can probably also be found similarly discounted from their new price.
Good luck with your family pictures.
Edit: saw that this is an old question that poped up, if you are still active you could let us know what you ended up getting in a comment