Tires and Brakes is what you need to be able to drive properly, skill of course needed in the meaning of properly. Brakes (pads, fluids) make sure you don't end up in the walls. Tires shave off the time most.
Here is a nice test done by Sport Auto magazine (in German) in Hockenheim, which just shows the advantage of semi-slicks in a Audi TT-S.
(Seriereifen = standard tires, Sportreifen = Semi-slicks, Fahrwerk = Suspension, Leistung = Chip-tuning)
Just with semi-slicks they got almost 2sec of the laptime, with more power (43 bhp) just 0,7sec.
So to agree on most of the comments, get some good tires and learn to drive
Edit: Ah, jumped to conclusions without really reading you post fully, like most. Seems you already have everything there done. Then power doesn't hurt. Supercharging if probably the only way to get real power, so LSD and better ratio there to suite your track might be the next steps. Shorter ratio will boost the acceleration nicely and LSD allows you to fully utilize that acceleration out-of-the-corner. Also make sure you have the set-up done correctly, corner weighting the coilovers and having proper wheel alignment for the semi-slicks.