$1800 and you are looking at the entry level cameras? May I ask why? (There could be a very valid reason)
By your screen name I am venturing a guess you are going to eventually want to do this on a more professional level...
You need to be aware of that and the longevity of your camera gear.
This is my mindset: EVERYTHING is in your lenses when it comes to the bottom line on image quality. However, you don't know what lenses you will want and need at this point. You don't knwo what your style of shooting these things will be; you know probably little about controlling the image yet; you also know very little about the levels of light and how your lenses will perform with them. Until you know what you don't know purchasing lenses is a shot in the dark. It usually is NOT what you want once you get a little knowledge under your belt. In which case it makes more sense to invest in the best, most up to date camera body you can purchase within your budget. You can then use that camera for a much longer time before it becomes a pinch and you need to upgrade. In doing so it allows you to put all future $ towards good lenses instead of constantly being behind the 8ball updating bodies. You have the budget to do it.
Both the T3i and the D5100 are entry level consumer grade DSLR's. NOTHING wrong with that, but they are what they are.
The D5100 also does not have an auto focus motor in the body. Meaning that when you purchase lenses you have to either purchase the lens with the motor in it-more money in each lens-or you have to manual focus only. It does limit you to only manual focus on a few pretty great lenses.
You also specifically mention sports down the road. However both of these cameras FPS is pretty slow for sports. The canon is 3.7FPS and the Nikon is 4.
They are also not weather sealed, so shooting outdoors in inclement conditions for sports is a concern. Not something that isn't easily gotten around, but still.
Stepping up to the 60D in canon offers you more FPS at 5.3, a much better focus system plus a better viewfinder and IQ improvement. 60D kit runs about $1100 AND it includes a zoom lens with a longer reach than the T3i-nice as you are learning because the first thing people discover they want is a little zoom. Still not weather sealed. Still a plastic body
Stepping up to the D7000 in Nikon brings you several improvements over the D5100 including a MUCH improved focus system, better viewfinder, 6 frames per second **built in auto focus motor***, 2 memory card slots and weather sealing. And the kit lens also has a bit more zoom in it as well. Kit runs about $1499. Ok, so that is a bigger step than the 60D was, but it's INCREDIBLY worth it.
ONE MORe option I want to put on the table with those is the Canon 7D. Again the 7D has the same improvements over the T3i as the 60D has PLUS weather sealing and 8 FPS. It's also handling high ISO's better which is a big deal when shooting sports-you WILL max out your ISO. Kit on that about $1560.