You've received some good advice. I agree that the Delta lathe at around $600 would be a good starting place.
If you're serious about the $1000 limit, for the time being forget about a chuck. You do NOT need a chuck to turn bowls. The lathe will come with a faceplate. Use that and glue blocks. Actually a more secure fixing than a 4-jaw chuck. I suggest that you buy a few quality tools instead of cheapies. Forget used tools. They'll probably be poor quality.
With what's left of the $ do yourself a very big a favor and get a lesson in basic woodturning. A good instructor will help you with tool techniques and more, but will also give you knowledgable advice on what tools to get to start out. Woodworking stores such as Woodcraft regularly have lessons available for not much money.