Without knowing really what genre you're interested in, I'll go ahead and suggest The Squier Classic Vibe line:
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I'll admit I can be a bit of a gear snob, but I have to say, that the Classic Line has been really impressive. They're the standards of bass, a nice place to start from as a ground work and until you figure out what direction you're going in. Either the 60s Jazz or Precision would serve you very well for a good while. On the other hand, I'm not sure what type of neck and such you're comfortable on, maybe you'd prefer a Mustang or another short scale instrument. Definitely agree with folks that shopping around, as it were, is a fantastic idea. Try lots of basses.
I agree on the better amp as well. It can make a serious difference. The first amp I had was a Fender Frontman 15R, or something like that. Little dinky guitar amp, in no way made for bass, but that's what I had for the first 9 months of playing. I eventually upgraded because I ended up toasting the speaker in the poor Frontman, got lucky and a family friend was offloading a beat up Ampeg B2R. It was a world of difference, and I sounded markedly better. So, you might also want to look into amps before you're set on a new bass (though trust me, I know the feeling of wanting to get another instrument, it can become a serious problem (I shouldn't be allowed in music stores with my wallet, I have no self-control)).
And finally, in closing, here's something else to chew on: instead of buying a new starter/mid-level bass now, start saving, wait a while longer, and get an even nicer bass. That's what I did. I paid my dues on this horrendous Jazz knock off. The thing buzzed and had a neck bow and flatwound strings like bridge cables, but instead of buying a new bass at the first opportunity I got, I set my sights on a Fender American Vintage 62 Precision bass. I saved for probably a year, allowance, yard work money, birthday, christmas money, whatever, but I finally bought the AVRI Precision as my first "real" bass, and that bass has been pretty much my undisputed #1 for going on 6 years (now in a tie with my new 62 AVRI Jazz bass) and that I'll probably keep for another 60 years if I can manage. So think about that: save up, and buy something even nicer that will last you even longer.
If you have any other bass or amp related questions, feel free to PM me. Remember, the search function is a brilliant tool while researching basses to find others opinions.
Oh, and welcome!