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 find something good for your purposes for $130 or less. Don't buy a "starter set".