I started out 17 years ago with a Delta 46-460 mini lathe (or something like that) and then a Jet 1014. I quickly became aware of its limitations and traded up to the Jet 1221-VS. Great machine! I wanted to also turn large pieces and purchased the Powermatic 3520C that I now do all of my turning on. I set up the Jet 1221-VS as a polishing and squaring station and that has worked out well for my little shop. I'll never sell it. It has run like a champ for years! I did upgrade the tailstock and banjo clamps with Rick Herrell's banjo clamp nuts, and that did the trick for making them both rock-solid. No more movement. Rick Herrell's squaring jig is also a 'must have'. In fact, pretty much everything he makes is a must have! (LOL!) Get as much horse power you can afford. You'll need it down the road for those bowls, boxes and vases, I promise.
My thoughts on 'budget' lathes, well...I'm not a fan of them. I've been down that road and personally, it only produced a lot of frustration, horrible run-out, noisy motors, squealing belts and loose tailstocks as well as crappy banjos and bed-ways that didn't weigh enough. Get the best you can afford with what you've saved. If it takes a little longer to save up for that Jet 1221-VS, be frugal and patient and squirrel away every penny you can for however long it takes. Figure in what you can sell your current lathe for, even if it's just for parts and scrap metal and put that towards paying for your new one. You won't regret it, ever! However, you will regret buying 'cheap'. Just my .02¢ worth of advice.