I suppose you could call me lucky(?) in that respect. I knew what a lathe was and roughly what it could do a long time before I got one. Now I have several, with more in my past. The big issue was more a place to keep it than cost. With that said, the first project might be something like a chess set. Brass and aluminium will give good contrast. Brass is expensive though. Make it all from aluminium and blacken one set. That's always a good project even if you don't play chess. Me, personally, I just play at it. But with so many pieces to make mistakes on, what more could you ask.
Then there are the little "tooling" ideas. You don't need something so complex you spend more time figuring it out than making it. Maybe a soft hammer... ... Or that punch you need to drive a nail just so. Most anything you want to buy can be built, given enough time. The opportunity to learn some new technique is wide open.
And finally, from long winded me anyway, is to repair something around the house. The lathe is just another tool, not the "end all" that many make it out to be. Albeit expensive, I will concede. You can turn most any material you want. Even water, though you have to freeze it first. Air now, it might be a problem... ...
Welcome to the hobby, my friend;
Bill Hudson