Yeah, actually, as you are seeing...you can get either a good Alto or Tenor for that $. You can also get a sh#t Alto or Tenor for that $.
Buyer's market these days, so I disagree (respectfully) with those who say you need to increase your budget for a Tenor. Landing a respectable Tenor which can last you forever (i.e. no need to later "buy up") is very, very feasible if you are willing to do a bit of research for yourself.
Any number of used horns can be had for that sorta dolla'...some of my favorites in this pricerange include:
~ Vito (Beaugnier/Kenosha horns...not the later Yamaha stencils)
~ Conn Director usa-made, or Pan Americans
~ Buescher 20A/30A or perhaps Aristocrat
~ Martin Indianas
~ Olds Ambassador Parisians (by Pierret, France)
~ King Clevelands
~ Holtons (Great sleepers)
~ Dorfler and Jorka stencils (Genet, Maure, DuMont, Roxy, Voss, etc.)
~ Keilwerth stencils from the 50's-'70's
~ Kohlert and their stencils, same era as Keilwerths
~ Noblets (again, by Beaugnier)
(the last 3 might be pu$hing it for a Tenor, but an Alto is definitely feasible)
If you don't play...do this:
1) go online, find some vids for "how to blow into a sax mouthpiece".
2) Go buy a sax mouthpiece or two...just a Yamaha 4c or 5c. They're cheap. Learn to blow it properly.
3) Go to a horn shop (or a school band room if you are a student...or maybe you have a buddy who plays sax) and try blowing some Altos and Tenors.
4) See which one moves you more..the Alto or the Tenor.
5) Take your $600 and buy a nice horn. Then take lessons.
It usually goes sumthin' like that.....oh....
6) Have Fun.