Based on the kind of music you prefer, a Fender style Jazz bass sounds like it would suit you best. Maybe even a Jaguar (great for guitarists switching over to bass).
Not to go into brands as you mentioned, but reasons for recommending the Jazz- thinner neck than the P Bass and the dual jazz pickups have more bite to them than the P-bass pickups which are..well, thumpier for lack of a better word. I've had quite a number of basses, active and passive- but really for old school type music, it's a no-brainer, I pick up my passive Fenders- none of my other basses can get the right kind of sound.
Another thing I have to mention - don't get a 30w amp. Even the best, most expensive bass will sound like crap on them- it'll probably discourage you if you hear what kind of sound comes out of it. The speaker is too small and the wattage too low to get crisp tones and sound. You'll want at least an 80w-150w even for playing around in the home. Easier to turn down the volume than being underpowered to begin with. A used amp on CL can be very reasonable and should easily fall within your budget. I bought my 150w Fender Bassman for $80.