As a fellow nooooooobie...
...let me offer my insight by building on bonsai's post:
1)
A Lathe, since you can't do any turning without one.
I started off by finding a 1953 ShopSmith ER10 at a garage sale for $175. I'm still using it to turn, and have done bowls up to 12" with it, with only a little bit of fear
I'm glad I was able to find something inexpensive to start with. This allowed me to find the direction I wanted to go (I've fallen in love with bowls), so I was able to make an educated decision when it came time to buy a big lathe.
You can get a Mini lathe for < $200 to get started. That'll let you do pens, ornaments, small bowls, goblets, boxes. Or, as other posters have mentioned, find a club so you can play with their machines for free.
2)
Shapening Equipment
Absolutely. I have a 6" Delta vari-speed grinder, for which I made my own Wolverine-style jig from King Heple's plans. Most people will recommend an 8" grinder. It's definately better, but not neccesary.
The important thing here is not to fear sharpening. Once you do it once or twice, sharpening touch-ups at the wheel only take a few seconds. And as bonsai said, a sharp tool makes a WORLD of difference. I was trying to figure out why my roughing gouge wasn't roughing. I changed my grip, changed my stance, presented the tool differently, nothing worked. Then the light bulb came on, I spent 30 seconds at the grinder, and the chips were a-flyin.
3)
Turning Tools, You are much better off with a few good quality tools than a bunch of middle of the road ones.
Well, I'll half agree here. While this is good advice, it's hard to know which tools you'll want to buy if you haven't used them. I picked up a $30 set of Buck Bros chisels at HD to start with, so I learned what a skew does (besides catch), how a gouge works, etc.
That said, if you've decided you just want to do bowls, a good 1/2" bowl gouge and perhaps a scraper would be all the basics you need. I use my bowl gouge almost exclusively when I turn bowls. For around $50, you can get a very high-quality bowl gouge that will last you a lifetime. I just purchased a Crown Pro PM 1/2" gouge from Tangboy5000 for somewhere around that price. I can't wait to play with it
Most importantly, BE SAFE AND ENJOY.
Hear, hear! Turning is absolutely the most fun I've had with a piece of wood, next to skateboarding
And be sure to post pictures of your work, both good and bad. It's always a learning experience.
OH! How could I forget? If you want to make bowls, grab Bill Grumbine's DVD. I read this advice over and over again in the forums, and said, "How good could it be?" Then I was fortunate enough to get a lession with Mr. G, and bought his DVD while I was there. Best $30 I've spent in my turning career. If nothing else, for the dry humor
-Joe