Hello, You could try to use a clay bar? I use Meguires quick detailer spray and their brand clay bar, but I found this video to show you. And since you just bought it, what a great way to get it all cleaned up and start with a fresh coat of wax
I hope this helps with the marks
Sounds like you aren't a detailer... I'd try clay bar then maybe one of the Poorboy remover products (they have a bird **** remover and some tar remover and some other products that are safe for the paint).