Get yourself a Fender Squire Precision bass - they run from $200 to $400 and are quite good quality - great bang for the buck. Even pros sometimes use them on stage (they swap out the logo) so that their high-end gear doesn't get banged around. It's a bass you can use for a long time.

For an amp, look into the Hartke practice amps. Also good quality and feature rich - you can plug in phones, an MP3 player, or practice through the speaker at decent volume/quality that won't drive your neighbors nuts. You can find something good for your purposes for $130 or less. 

**Important:** If you plan on practicing through phones (you inevitably will end up doing that unless you live alone, surrounded by a decent amount of empty space) you need a good set of phones to handle the lower registers of your bass. Cheap phones will fart, get blown out, sound horrible...

You should also get yourself a metronome, if you don't already have one, or better still, a little rhythm track generator. Korg makes a portable one with its own speaker that works on batteries for about $75. Bass is rhythm instrument - a timekeeper - and works with the drummer, so that's extremely important to have.

Also, don't buy a "starter set" and don't buy one of those amps with a zillion effects and noises, unless you intend on spending a few hundred $ for a Roland. Your goal right now is to get a good playing and sounding bass and a nice clean, deep sound out of your amp.