The 600d or any crop camera takes awesome pictures. A FF camera only main benefit is better lower noise at high Iso and thin dof shots. If you view random pictures in this forum, 95% of them, most people cant tell if they are taken on crop or FF.
Best way to improve picture quality is to shot in raw and use post production. Best way to improve qaulity of pictures is getting better by taking some more pictures.
I don't agree with your assessment. If the only place your images end up on is the web...then yeh there will be very little difference between crops and FF cameras. But if you make prints...then there will be a large difference, at least from my own testing and expereience using a 7d and 5d2. Prints unto say 11x14 are very hard to see the differences...but anything larger shows up in much better fine details and tonal qualities in the FF photos.
If you shoot just for the web, today's advanced P&S like the new Sony is all you really need.