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.
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. If you're planning on patching in computer or tablet audio, you can get all sorts of software and apps for rhythms that are free, or very low cost.
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...
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.
If you want to spend around $300, you can get a Roland Micro-Cube amp, which is worth every penny you pay for it: It is excellent quality, small, light and portable (can even run on batteries), produces an excellent bass sound at low volumes, includes a very good rhythm machine and some special effects that work well. Of course it also supports phones and an MP3 player going in. It's a great all in one solution - but for low volume only. If that's too much to spend, go with a Hartke.