I started on the 600D as well.
I can say it's a fantastic camera, I really loved it.
It has great features and a bunch of auto settings to get you up and running quickly, though I'd advise to get into manual as soon as you can.
The image quality is superb, on par with the 7D, which is what I currently use.
In my opinion you can stick with the kit lens while you're learning, unless you really have a specific idea and plan what you want to shoot.
I know lot of people will say "get a new lens, get a new lens"... but if you're just starting out and want to find your way in photography, save your money, stick with the kit lens for a while until you get an idea of the type of subjects you want to shoot.
You can get great images with the kit lens imo.
If anything, get an off camera flash first. It will really be a big help, plus the 600D has built in wireless flash control. I used that feature a lot and you can get very creative once you have a flash.
The screen is great too, I didn't do a lot of video shooting but I love shooting pictures from a low angle, close to the ground and the flip out screen really helps there.
Overall I can't fault the camera and I probably would have kept shooting with it, had I not purchased a 24-70L lens, which attached to the 600D made it front heavy and unbalanced in my hands.
There is a 600D owners thread on here somewhere, have a look at it if you haven't already.
Good luck and enjoy!