OK, you're confusing a couple things:
1) Resolution, as in the physical resolution of the video, does not get changed or impacted by the lens whatsoever. You set your camera body to 1920x1080 (1080p) or 1280x720 or whatever and that's it. Changing the lens will NOT affect that in any way.
2) Graininess, which is an aspect of the quality of the video, not the resolution, can be caused by any number of things. Some of those ARE impacted by the lens. Generally, graininess is a result of poor lighting or not enough light reaching the lens. Better lenses require less light to work and do better with good light than a lower quality lens. If you have the stock 18-55 f3.6-f.5.4 lens, you can get very good results but will need enough light. And that light needs to be in a place where the lens can actually use it. Without seeing samples of your videos, I can't comment much further other than to say lighting is by FAR the #1 item to look for when fixing image issues (assuming focus isn't an issue).
3) Your Canon T3i will do 1080p @ 24 frames per second and 720p @ 60fps. Both are fine. Personally, for stuff like you're doing, I would rather have the 60fps. If you're doing solves with fast turning, you want to have a high frame rate to help capture the movement smoothly. 720p isn't the greatest resolution but it still looks good and is the resolution that most cable television is broadcast in, so people are watching it on big screen TVs without an issue.
If you've got a sample, I'd be happy to take a look at it and offer suggestions. JRCuber has a video showing his setup and his videos, especially his most recent stuff, is well-lit and nice quality. Check it out:
