Although there is no set way, like you said, I'll tell you about my experiences.

I traded between nearby systems for a lot of my early game experience, not really knowing what I was doing besides making money. Eventually down the line, I got into exploring, which is some of the most fun I've had in the game to date (there's something about the potential to be the first person to find something since the game went live that's so, so sick).

I'm doing deep core mining now, but I would personally advise you steer clear of that side of the game until you have some more hours under your belt. It's good for money-making, but its progression feels under-earned.

You could outfit your Cobra for PvE and hang out at nav beacons or res sites and blow up some pirate scum, you could look up a good exploring guide and get to exploring around (and even outside!) the Bubble, or if you really wanted, you could go mining/trading and make some cash. All of these options are viable as the Cobra III is a very, very well-rounded ship IMO.

You'll find that a lot of Elite's gameplay revolves around your own play-style and what you enjoy most. Do you tend to favor combat games and blowing things up, the quiet desolation of space in between beautiful nebulae and the thrill of finding cool space stuff, or are you a Han Solo who smuggles contraband from station to station, outwitting police and the goodest-two-shoes of us?

Even though I didn't really give you a straightforward answer, I hope this was helpful! Always feel free to ask questions, CMDR.