As far as attachments for your lathe go... Here's how that shook out for me:
- I started with a VERY primitive lathe, the first thing I bought was a center finder, a 1/2" spindle gouge, a parting tool, and a full face shield.
- Then about 5 minutes later you figure out you need to be able to sharpen your tools. I already had a Tormek, but needed the attachments to sharpen turning tools.
- Since everyone seems to be afraid of the skew chisel I bought one and started practicing. Then comes the skew chisel sharpening attachment.
At this point you've got the makings of lots of weekends goofing around entertaining yourself. Take a piece of firewood, chuck it into the lathe, make it round, then make some shapes. Don't worry about making anything more specific than a big mess.
At this point your priorities become brushes and brooms to clean up the mess you had a ball making, plus a woodturners smock when you realize the shavings managed to get down the collar of your shirt and your wife keeps yelling at you for tracking shavings all over the house (especially the path between your workshop and the bathroom). Also pick up a block of carnuba and you can play with putting a quick finish on the pieces of firewood you become most proud of.
Things that come a little later are a roughing gouge (you just took it real slow with the spindle gouge before and were fine), then someone tells you it's cool to make pens so you get a pen mandrel, some other pen supplies, and a simple pen kit. Somewhere along the way you see a Youtube video of a guy making snowmen on the lathe and decide that's good practice making large beads.
The best advice I saw was a picture of a bunch of spindles with lots of beads cut in and the caption was "if you're a woodturner just starting out, this is what all your firewood should look like".