A lot of your success will be determined mainly by you- and your measure of what success is. Really though, you should primarily work on getting some better lenses initially- but also - be stern in a commitment to using your camera routinely. For example, the 50mm f/1.4 lens would be a good starting point to start with- be sure to stop it down some for better sharpness- for example- but it would make a fine lens to start out with. this would have been the end of my post- had I not looked at the actual specs of the t3.
OK, not a fan of the T3 model- but I am sure that it's got limits just like each and every other model out there. The T3 is handicapped to some degree- more so than the t3i- which is kind of handicapped when compared to a better camera like the 7D. When you start running into the limits- is when you should consider upgrading the body... for instance, your center AF sensor is f/5.6 sensitive- which means that as the t3's best AF sensor- it is only as good as the worst sensor on the 7D !. In normal daylight- this is not necessarily going to be a limitation- but it certainly will be in lower light. Question: Am I being too blunt here or is it better to know what the limits of this camera is before saying that it cannot be the body ? I think not. Let's face it- slapping any lens such as a f/2.8 lens on it and thinking it will focus better- is probably a pipe dream. Everyone out here that has had a better body generally knows that the standard center cross type af sensor is generally the most accurate- and the one in the T3 is only as sensitive as the outer points on a t4i or 7d or 60D. You also have a MP limitation.. If you are framing your pics the way you want- without having to resort to heavier cropping- then 12 MP should be more than enough. If you find yourself cropping, then having a higher MP will help to a degree.. The kit lens- 18-55mm some love it- some hate it. Quality wise- it's OK- but that is about it. Stop it down some- and it's a bit better- but it's still just a cheap kit lens.
Really, if it were me- the first thing I would do would be to swap out the body. But I would want to know how to "cheat".. For example, the 7D is an excellent camera with an established reputation. It's a sporting camera- meant to capture action. It's got a sophisticated AF system- probably more than what most people need- but one of the key points of its AF system is the All cross type AF points- which helps when tracking fast moving subjects. The t3i, for example, in contrast has a 9 point, cross point in the center only- with the center being f/2.8 sensitive. the other AF points on the t3i are sensitive to f/5.6, and only sensitive to contrast in a direction perpendicular to the AF sensor. Even with a t3i, and its center point- being more sensitive- you will get better and more consistent focusing in lower light- and when using it for tracking in AI servo- using just the center point- it can do a fairly good job..
The only other crop I would consider- would be the t4i- as it has all 9 cross type AF sensors- with the center f/2.8 sensitive. It's AF system is not as sophisticated as the 7D, but it is cheaper, lighter, has a swivel screen useful for a number of things, and is generally good enough for most starting out- not too complicated- but not to simplistic where you reach its limits too quickly either. The lens to get with this camera is the 18-135 IS STM lens- puts the 18-55 IS to shame, IMO.
The decision to replace the body for you should really focus on where its weaknesses lay- and whether or not you will be attempting to shoot where it's weakest- i.e. hitting its limits.. From the specs, the t3 is more of a daylight camera - more suited for bright conditions. If that's what you shoot in mostly- use it - you already have it.. Get better glass- for image quality, DOF, bokeh, etc.. If you shoot and start hitting, say - 1600 & 3200 ISO in order to achieve decent shutter speeds- it's probably time to replace it.. I think at that point- your body is handicapped..
Current cameras, I would recommend;
7D.. T4i with the 18-135 STM lens (certainly better than the 18-55 IS ). I'd go with these because the 7D is the best crop camera Canon makes for sports (tracking, etc.. ). The t4i is a lower cost alternative- packed with features.. The 60D is an alternative- but in a lot of ways- the t4i is a mini-60D+, with the t4i being a newer model- I'd take that over the older 60D.
Lenses I would consider.. 50mm f/1.4 - certainly. A prime is good for everyone to at least try.. Different.. Canon 15-85 (mainly because of your camera's tending towards daylight anyways) - certainly. You would be able to use this and know that it's not your equipment holding you back under most cases.
Other things; Canon 420 EX II for an external flash... Whole new world once you get an external flash- best investment you can make- period.
Beyond that- I'd really say that your body is too limited to NOT think that it is not a limiting factor- at least in some respects.
That's my opinion.
EOS 5D III, EOS 7D,EOS Rebel T4i, Canon 70-200 f/2.8 IS II, Canon 24-105L, Canon 18-135 IS STM, 1.4x TC III, 2.0x TC III, Σ 50mm f/1.4, Σ 17-50 OS, Σ 70-200 OS, Σ 50-500 OS, Σ 1.4x TC, Σ 2.0x TC, 580EXII(3), Canon SX-40, Canon S100
Fond memories: Rebel T1i, Canon 18-55 IS, Canon 55-250 IS, 18-135 IS (Given to a good home)...