I originally wanted to write about 3 paragraphs worth of advice from my own personal experience of trial and error, but instead, I'll just lend you a condensed version of my 2-cents on recommended gear.
For an amp, buy a small practice combo, and make it the Acoustic B20 bundle. At Music123.com they sell the bundle for $160 - no tax & free shipping. It comes with the amp, strap, gig bag, instructional DVD, polisher, guitar rag, string winder, and a tuner. It's 12" speaker is large enough for a nice low-volume jam and it comes with a headphone and MP3 aux jack. Acoustic brand amps are fantastic for the price and sound to boot.
Link Removed
As for a bass, I recommend Squier. You can easily find a Squier Affinity Precision bass on Craigslist for $100 or you can expand your budget for a Vintage Modified Precision bass. I would recommend saving up the extra for one of those, but in the end, it's your choice.
Last, but not least, try out them Fender Light Gauge Flatwounds! They're only $19! Play with the stock strings (most will have some sort of rounds on them) and then spend under $20 to get you some strings that can bring out the "thump"! ...But that's just me.