First off, great to see you here. Hope the studies and training are going well. ( I am sure they are)
All lifts need the proper shoes. Running/cross trainers will crush as the weight increases (and I know you are well past that point). Balance will be off.
Olympic lifts are technique and explosive power. (and strength obviously)
Ex:
http://www.dynamic-eleiko.com/products/shoesFR.html The Ironwork II are very popular. The heal adds stability and balance to the lift and along with the arch will give you better starting position. The rubber on the bottom front will prevent sliding when you explode. Some squatter who squat narrow like the heal as it will allow them to reach proper depth easier.
Get some Chuck Taylors. I know thy are not sylish but they do make them in all black.

Great for SQ because they are flat and allow you to sit back on your heal while SQing. Balance and form will be better and weights will increase. You can also use them for DL. I prefer wrestling shoes but... Either one will prevent you from leaning over or getting pulled from the shoes collapsing and allows you to drive with your heal.