You can do Elite runs in a free 9-console ship loaded with Mk8 gear (though better hardware makes it easier obviously, and Mk11 is cheap so no reason not to). Tactical awareness and a reasonable setup is much more important (doesn't have to be amazing, merely reasonable). Reputation gear and fleet ships are cool toys that make it easier, but aren't a requirement, plus if you're driving a Vesta then you're already at Fleet-level anyways.
Its clich? to say, but your brain really is the best weapon in ESTFs. Once you know the Borg patterns and how to exploit them, they become pretty easy. Really the challenge is not to win (which will be a given), but to do better and better as you win and to figure out the best setups for doing so.
I've only recently started doing STFs, but here's what I've picked up so far:
1. Anti-protons seem to be the weapons of choice for the crit damage and since the borg appear to shrug off the procs of other weapons.
2. Resists to plasma damage in the shields, like the MACO shield, and perhaps armor. A BO with at least one copy of hazard emitters.
3. STO Academy has some walkthroughs on their site - http://www.stoacademy.com/walkthrough/stf/