Jeff didn't mention that turning is addictive.....
One important bit if advice. Do not rush! Take your time, visit clubs. Most have mentoring programs where you can visit with members see what they are doing, and what type of equipment they use, and sometimes even turn on various different machines. Take beginner classes at WoodCraft (or other stores) and make sure that turning is what you want to do.
Another is to look at your budget. The lathe is a big item, but not the only thing you need. An inexpensive set of tools will run $100+, calipers and other measuring devices $50+, face shield and dust protection $50+++, grinder for sharpening tools $75+. Sandpaper & finishing supplies. That's for starters. You will probably want to add a chuck $100+. Oh yeah, almost forget about dust control, a shop vac or a dust collection system is a must unless you are turning outside under an old elm tree. Don't buy "cheap" equipment, there are many inexpensive but quality products out there, or even used. Likewise don't over buy. You don't know if turning really is for you. If you are thinking of a used lathe, ask an experienced turner to go with you to see if is worth what is being asked.
If you buy a mini or midi lathe, you will need to build a bench with a fairly heavy top to help absorb vibration, unless you are only turning pens.
You will probably want to add grinding jigs like the Wolverine set from OneWay until you learn to grind your tools freehand. I didn't know how bad my freehand grinding was until I got a set and used freshly sharpened tools off the jigs.
I could go on, but I am sure others will jump in with other suggestions and even contradict each other.
JimQ