Depends on how much volume you need... are you playing metal? jazz? classic rock? country? noise rock? English music hall?
For $600, which is a decent starter budget, you want to balance how much volume you'll need versus the best available tone.
First, if you check the FAQ in this forum it can give you some good tips for what to upgrade first and what to spend most of your money on. Second, try to figure out if there are any bassists that you want to sound like and try to figure out what gear they play or how best to get close to that sound. Third, when dealing with a budget, a good idea is to go used... many dependable, decent heads and cabs (because I think you'll want to go with separate heads and cabs versus all-in-one combos) can be had for good prices used.
$600 isn't a lot, but you can get a decent (and loud, if you need it) setup... Peavey, Hartke, Ashdown all make good heads that are easy to find (Peavey's Nitrobass , Firebass, and MAX line, Hartke's 3500, and Ashdown's MAG series)... Gallien-Krueger older heads sound great and can be found used for pretty cheap (the 800RB and 400RB). For cabs, you can find a used Peavey 4x10 or 1x15 (depending on your needs and preferances). Also, lots of people recommend Avatar cabs (avatarspeakers.com) for new, inexpensive cabs.
I'm sure others can chime in with other good suggestions... it's place to start at least. Again be sure to check the FAQ of this forum and use the search function to learn more about specific amps and setups that you're interested in.