For bounty hunting I spent a long time in an eagle to begin with and found it to be pretty effective. It is really manoeuvrable so once you've got the hang of handling it you can take on much bigger class ships.
It's worth getting a kill warrant scanner if you are bounty hunting, as it will allow you to bag bigger bounties from other systems - without it you only get the bounties from the system you are currently in. As for other kit, it's a bit of trial and error to see what works for you - obviously bigger guns are better, and a mix of thermal to take down shields and kinetic to rip through the hull once the shields are down is good. I prefer beam lasers and multi-cannons, but everyone will have heir own ideas. Gimbled weapons will also make a big difference in hit rate over fixed unless you have much better targeting control that I do!
Shield cell banks may also be useful as the eagle has a pretty flimsy hull so you want to keep your shields up as much as possible.
As you get more credits, try to work towards getting a Viper if you want to stay bounty hunting.
No idea about mining I'm afraid as I've not tried that.
Good luck!