Well, what bass to go for is a pretty personal choice. If you have no idea, it’s not the worst idea to just go with what appeals to you visually among whatever basses you can afford. Or something similar to what one of your favorite players plays, or what a friend recommends, etc. Me, I like the classics: the Fender Precision, Fender Jazz and also the short scale Fender Mustang, plus I’ve always thought the Gibson Thunderbird is cool. Less expensive versions of those Fender models are made by Squire and less expensive versions of Gibson basses are made by Epiphone (plus Epiphone has some original basses not made by Gibson, such as the Allen Woody Rumblekat).
In bass-speak, "P" stands for Precision and "J" stands for Jazz. A PJ model looks like a Precision but includes, along with the P pickup, a J pickup for some more variety. I mention this because the Squire Affinity range has some PJ models that could fit in your budget straight up (sans amp), but bought used would leave you with some money left for your ampage.
Amp-wise, you could get almost anything for a small practice amp to learn to play on at home. For stage use, well, it depends on how big a stage you’ll be playing, the PA support and how much you’re willing to carry around, but assuming you’ll buy a solid state amp, I’d say you want something that puts out a minimum of about 200 Watts, preferably double that for playing with a drummer. I use a 500w amp, which is actually more than I need. My smaller amp is 150 Watts and that is good for "tasteful" type stuff or restrained band practice (and much more than enough for home practice), but insufficient for rocking out with a hard hitting drummer and loud guitarist. Others can recommend models better than I as I haven’t bought a new amp in a quite a while now.
What else do you need? Well, if you buy new, and sometimes even used, you won’t necessaily need new strings. As bass strings are expensive, that helps! You’ll want what’s called
a "setup", making sure the neck is set right and the strings are set to the right height and that the notes play in the proper pitch. This is pretty important since if your bass isn’t set up properly, that will lead to all sorts of struggles (and frustration) that shouldn’t be there and that aren’t the player’s fault. Most (all?) music shops have someone called a "tech" or "guitar tech" who can do the setup for a not-too-large fee comparable to the price of a set of strings. If you buy your bass from a shop, try to get them to include a setup for no additional cost. Finally you’ll need
an instrument cable or two (2nd one for backup or for letting you place a pedal in the middle) and
a tuner. For rock, many players use
a pick (plectrum), so you might as well pick up 2 or 3 of those. Some people only play with fingers, but picks aren’t expensive and you might as well have one or two around (I said 3 above because you’ll quickly lose one
) to see if you get on with them.
Where possible, buy used to make your money go farther!
Hope all that helps. Good luck and enjoy the journey!