DO NOT RUN A BASS THROUGH A GUITAR AMP!
you WILL damage the amp sooner rather than later by doing this.
there are a bunch of factors that will affect your tone. strings, EQ configuration, amp manufacturer, effect pedals, etc.
i find new bass strings sound like crap, especially round wound strings (the most common). they're really bright and metallic until you wear off the first bit of shine. flat wound (or ribbon wound) strings generally cost more but sound a lot smoother.
lower quality amps will also have a narrower range for their EQ settings. play around with it until it sounds decent, don't just set the knobs to what you think it should look like or what you saw in someone else's setup diagram in a magazine. again a student level practice amp is not going to offer the same range and response as a combo amp geared for an intermediate player looking to play their first gigs.
you could also try cutting the tone using the knobs on your bass and stick to playing with the neck pickup for a smoother sound.
as for models in your price range there's a bunch of good quality stuff for between $200-$300 out there.
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i would definitely take a closer look at the ampeg or the gallien krueger combo amps, peavey is also solid too. there's also a smaller version of the warwick amp that i've been using for the last year or so which provides great low end without getting muddy (and that's with a 5 string).
either way you've got options in your price range. don't bother with cabinets yet, as you would also need to invest in a head. a good combo amp should do the trick.