Personally speaking, I'm a big fan of basses with a Precision/Jazz pickup configuration. My first two were P/J's…a Fender Aerodyne (I lucked out on a great used one) and a Squier P-Bass Special (a hybrid P/J, both in electronics and neck/body). Although they don't sound exactly like their "pure-blooded" cousins, they have a great range of tones at your disposal.
Now the former may be a bit beyond your price range (like I said, I lucked out on my Aerodyne…$376 shipped!) and the latter has since been discontinued, but if this sounds like an interesting prospect, Squier just unveiled a pair of Precision PJ's…an Affinity and a Vintage Modified.
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If you prefer a Jazz-style bass, the Classic Vibe and Vintage Modified's get a lot of love 'round these parts. (I daresay the CV gets a bit more props here) I can't weigh in much on Precisions…I prefer the slimmer Jazz necks, but that's just me.
Sadly, I can't weigh in on amps. I live in a place where that level of volume would be…unwelcome. To that effect, I get by on a Pocket RockIt headphone amp. But when I first started shopping for gear, I did notice Squier's Affinity Pack which comes with an itty-bitty Fender amp. Anyone here know if they're worth a damn?
Oh, on the topic of shopping…some people here poo-poo Guitar Center, and I'm certain it's not all without good reason. Any big-ticket retail center will have good and bad employees. For example, the one by is full of awesome guys who are pretty knowledgeable, but don't try to B.S. me if they don't know something. More importantly, they have a pretty darn good return policy, so if an axe isn't working out for you, you have 30(?) days to bring it back.
I hope this is a bit helpful, or at the very least, opens up some doors of thought. Happy Hunting!
EDIT: +1 to everything wcoffey just said right above me!