Weather you use Tor or VPN you want to use a different browser than you use for unprotected browsing. Better yet use a different machine.
It's the point what you're trying to mask - you mentioned browser and machine, so the identifiers that I can think up are: User Agent, plugins, encoding, language, etc. and the MAC (media access control) of the machine. The probability of exposed serials of hardware is extremely low, but theoretically possible. For both there is a more simple solution than physically changing the ground - which is also fine, but can be time expensive. For Firefox there is an addon called FireGloves, which deletes/nullifies all that things like UA, plugin list, language, encoding and other stuff. For changing the MAC there is plenty of MAC changing software - TMAC (even if it's not open-source and there are many open-source MAC changers available) worked fine for me. And also, TOR software generally comes bundled with a preconfigured Firefox, together called TBB (Tor Browsing Bundle).