In the USA, good luck, other places there is still a good chance. We still have clients that combine sites like localbitcoins and trading for the exit strategy, but believe what you want, there is little that is anonymous in the real exchange of FIAT to BTC and back again. Despite what you feel and believe, there are eyes on you every minute of the day, and they know what you do and do not do.
If you want to keep yourself anonymous while remaining in the world of trading, then I believe it is just your dream which can never be fulfilled. You actually need to reveal your identity and if you do not reveal it, even then you will be exposed.
There is no point of doing trade and that too keeping yourself anonymous. If you want to get into trading, then you have to get yourself ready for all the things.