Divi996TT- Welcome to Autopia!
Sorry to hear about the marring on your TT. It's hard to say whether your wash uncovered marring that was already there (but hidden by the dealer using a glaze/etc.) or whether you put the marks in while washing (washing without marring is much harder than people think). Probably a little of both.
To remove the marring you'll have to use an abrasive polish. And, sorry to say, it won't be easy as Audis have *very* hard clear (I have two at present and have had maybe a dozen total). This is something a layman *can* do, but it'll take a bit of research to learn how to do it and what stuff you'll need.
You could try some Meguiar's Scratch-X and work on the marring by hand, but IMO that'd take forever and a day and wouldn't work all that well anyhow. You could try some other product by hand (and I bet somebody will chime in with a suggestion or two), but IMO it'll almost certainly need to be polished by machine, so you'll have to either get a polisher/pads/polish or hire a (hopefully competent) pro to do it. I think that the short-term answer is to hire the pro for now and the long-term answer is to research this whole detailing thing so you can take care of it after the pro fixes the current issues.
Try searching on "foamgun" or look in the Hall of Fame forum here and check out my Non-marring Wash Technique. The toughest part of owning a black car is washing it without marring the paint. It's virtually impossible to do using "normal" wash techniques.
Hope my "find a pro" and "do a search" don't sound like blow-off responses, but fixing a black car's finish and keeping it nice is something I can't really cover in one post.
See what others post, I know of a few people here with black Audis.