Welcome Commander! I too love simply exploring and cruising around the galaxy taking in the sights. If bounty hunting is what you want to do then I'd suggest buying either an Eagle (good but cheap fighter) or a Viper (better but more costly fighter). The fastest way to get the credits for either of those is by trading. Space trucking. Buy goods in medium or high supply somewhere, take them to where you can sell them at medium or high demand, repeat, profit. Then once in your fighter use the bounty credits to slowly upgrade your ship, or keep space trucking to gain funds for upgrades faster.
I wanted to explore in an Asp from the outset, so I worked my way up the hauling ships by trading like so:
Sidewinder -> Adder -> Cobra -> Type 6 -> Asp
Once I had the Asp I space trucked a bit more to buy a few decent upgrades, and now I'm out in deep space exploring. An Eagle is cheap but a fun ship, wouldn't take you long to get it. The Viper takes a bit longer but isn't bad either. The galaxy is your oyster, enjoy!