I could record one track, then go back and record another on top of it and so on.
You are in luck because the feature is common even on cheap toy-like keyboards that seel for under $200.
Next question is what kind of quality you are after.
Also if you are willing to connect your keyboard to a computer then almost any keyboard can do this at a fairly sophisticated level.
What kinds of sounds do you need and is realism important. How many keys wide. Ranges are from small 25 key desktop keyboards up to full 88 keys.
No weighted keys and up to $600?? Look at the Casio WK500. It's about $300 or one of the Yamaha PSR series.
$600 is actually a no man's land. there are more to choose from at half that price or at four times the price. If there is a Guitar Center ior Sam Ash store near you. Go there and bring headphone and plan to stay a couple hours. Try out a few.
I'd suggest buying in thr $200 to $300 range then later when you know what you want making to jump to the profesional level like the Tyros, Pantom, Motif and so onbut those are low to mid four digit prices.