Would I effectively cover up my trace if I used a service like coinbase to transfer funds from my bank to the wallet of bitcoins on coinbase, then to an anonymous wallet like blockchain, and then from there to wherever i want, anonymously?
Everything involving your bank in traceable.
But blockchain is supposed to make it anonymous, and then if i sent it to another anon wallet it would be even more anonymous... shouldn't that cancel out the fact the bitcoins came from my bank account by the time i spend them?
No, bitcoin is not anonymous, it's just incorrectly advertised as such. If one bitcoin address gets tied to your identity you should assume every other is also.