Stock S3 will take Stage 1 remap with no other mods needed. 310hp
Stage 2 needs Cold air intake (CIA) + exhaust down pipe/sports cat + remap 320-330hp
Stage 2+ needs Stage 2 + high pressure fuel pump (HPFP) + remap 340-360hp
Most go Stage 1 then straight to 2+
A stock S3 DSG will take 2+ without changing clutch, but I think manuals need uprated clutch to go this far. Suspension and brake upgrades are recommended as the power goes up so get your wallet out.
Over 2+ and you getting into hybrid or big turbos and other engine mods.
All depends on your budget, I'm no expert but there is a lot of good info on here. Just read some other build threads for some ideas on how far you want to go.