Hi Sarah 2013 tt,
Congrats on your new car. I recently purchased an A3 S-line and I find the ride also very harsh and its getting on a bit now so definitely need an upgrade. After doing some research I also have decided on getting Bilstein B4's as they seem to be the best after reading reviews from other Audi owners. The upgraded springs that match the S-line models as they are lowered the same (30mm) are Eibach springs as the ride is much better. I will also be adding a powerflex rear anti-roll bar which firms up the rear slightly but reduces understeer. I imagine a TT sits nicer on the road anyway than the A3.
Let us know what you decide to go for!
Cheers,
Denver