Basically,
RoJoHen, you're looking for an
Elite-style game (a subgenre of space combat games named after the BBC Micro game that started it all

). Lots of great examples have been given in this thread already.
There's the
X-series. I've only played
X3: Terran Conflict. It's a really big universe and you fly around in a personal starship. You can buy other ships, upgrade current ships, engage in ship-to-ship combat accept a variety of missions (protect stations, kill a target, deliver x-number of goods somewhere, transport people to locations, etc), purchase stations and set up your own little empire. It's a pretty neat game, but you have to be patient early on. It has a fairly steep learning curve, and it starts off really slow. You're generally not put in a very good ship and there's not a lot that you'll be able to do successfully. But if you can hang in there past the beginning, you'll get better ships that will allow you to do more and things will go a lot better.
They're also coming out with a new one later this year,
X Rebirth, and they're apparently rebuilding the whole game engine from the ground up.
I think I introduced you to the X series back when X3TC came out.

I could talk about this series all day...
The games in that series all have a big learning curve, although not as bad as they used to be. The first game,
X - Beyond the Frontier, literally threw you in at the deep end with a bare bones ship with no weapons or time accelerator mode, and a mere 100 credits (and a debt to pay off later too!), and an unknown universe to explore painstakingly. It could take several hours - even days - to just get on your own feet at first, but once you do, the rest comes very naturally. There weren't even missions to do in that first game, plus you were stuck with the one ship to fly, so no flying others (although you could hire freelance wingmen by talking to them if friendly enough, and of course hire large transports to carry the stations you buy) and no capturing and selling enemy ships.
Thankfully the later games gave you better starts, a gradually bigger universe, more gameplay options including buying and flying capital ships, a much more detailed manual to read, and of course much tougher enemies.
In a way, depending on how far you progress, I think the later games (and in particular the later stages of those games) can be played either as an
Elite-style game, or a business simulation game, or even a
Homeworld-style real-time strategy - although in the latter two styles the interface is not at all a patch on the real things, and do take the series away from its
Elite-clone roots, despite it being an example of a logical progression of how such games would have developed. Maybe the series reboot, Rebirth, may do something with this. Or maybe not.
But yeah. Flying around the Universe. Helping some people and destroying others. Earning cash by scavenging discarded goods. Being a space taxi driver, or even a space bus driver. Building and living in your own station. Starting a trade empire and a goods haulage company. Prospecting and mining asteroids. Working for the police. Killing for fun and profit. Piracy and murder and making a quick getaway. Doing missions with interesting rewards. Stealing ships and selling the pilots to slavery. Leading your own privately-funded invasion fleet against the pirates. Good times.
