If you wanna shoot video with it, go for something newer. You might even want to go all the way up to the T4i (which I believe is the first DSLR to have full time autofocus in video mode). For artsy videos where you can pre-focus, the T3i is fine but if you wanna record your kids on the move, you'll probably want the T4i. (Be aware that DSLRs keep a much smaller depth range in focus - unless you to sacrifice a lot of light - than a compact point-and-shoot, so accurate focus becomes more critical.) But for taking photos, the T2i likely meets your needs.
For truly good action / low light performance, you'll need to spend a lot of money. The features that make it happen (big sensors, long zoom [?], and large apertures) are the things that are pricey.
If you're planning to shoot, say, indoor soccer, with a T2i and a kit lens, you might do better than with a point-and-shoot but you're still doing to struggle. Even moreso if you're planning to shoot video.
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