I did all the training missions and spent a few hours getting my controls perfect. Then I launched the game proper and I'm just plopped into a station. Is there a story to follow or do I just wander around trading from outpost to outpost? My only experience in space sims is Freespace 1&2, those had pretty cut and dry campaign missions.

There's no story in this game. It's not exactly a "make your own fun" kind of game, but it's a bit closer to that end of the spectrum than, say, Rogue Squadron. You should read about some of the different professions/roles you can follow in the game here. I'd suggest sticking to non-combat (and probably non-illegal) roles at first, as the starting ship isn't great at much. Play around with your options, get a feel for what you think you want to be doing, then set your first goal: which ship to upgrade to first.

I've decided to focus on exploring for now, and was able to upgrade to an Adder earlier this week. My next goal is to save up for a Detailed Surface Scanner, which will allow me to make money a lot more quickly while exploring.

Do some reading on what ships and upgrades there are in the game, and do a bit of planning to figure out what you want to work towards. There are a bunch of options and flexibility in what to do.

I wondered the same thing. I was completely lost and pretty put-off by the game. I did some more research (youtube, reddit, etc) and figured out quite a bit. I'd say the first thing to do is to just look for random bounty's to collect. Start 'Super Cruising' at a relatively low speed and search your navigation menu for 'Unidentified Signal Source.' These are basically random events (could be stolen cargo to be picked up, a wedding escort, or a fire fight going down between police and wanted criminals.) Target ships and see if they're 'WANTED'--if they are, kill 'em and collect the bounty. Just warp out when you're done and look for another Unidentified Signal Source. This will get you some starting cash. From there there's basically four 'professions' you can pursue: bounty hunting, trading, mining, and exploration.

Keep in mind that I believe you get a 100% sell back value on weapons/modules and a 90% sell back value on ships. I'm still very new to the game but I already have a fair share of fantastic stories to tell. The goal is make more money to buy better stuff; however you do it is up to you. How you get there, is kind of the narrative / fun of the game. I was pretty skeptical of the game from the Quick Look, but the amount of nostalgia it made me feel was enough to make me pick up a Saitek X52 and buy the game!