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.
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.