The Precision is a classic and although its limited in what sounds it can get it's certainly going to nail the ones you need. The jazz bass does have slightly more variety and looks a lot slicker, but personally I prefer the Precision. Try them out more with the neck in mind, in order to see which one you find the most comfortable. Honestly, the difference in sound is pretty minimal. And while I've never played one, I've heard that the Squire 'Classic Vibe' series is better than the MIM Fenders too.
As for amps, Warwick do a really good range of small combos in their
'Sweet' range. They have a wide range of tones and are very robust. I'd definitely recommend this if you're thinking of playing with a drummer, even if its just jamming (they come in 150 and 250 watt versions). If you're literally just going to be playing alone, or into a multitrack, then Warwick's
Blue Cab range offers a slightly less powerful (from 15-60 watts) but even more affordable option. Although both are incredibly cheap for what they offer. (New, the 'Sweet' range is around 300 euros, whereas the Blue Cab is about half that.) The only problem with either though is that you can't connect them to an extension cab - although given what you're looking for, that shouldn't be an issue.
Thomann Links:
Sweet 15:2:
http://www.thomann.de/fr/warwick_sweet_152.htm
Blue Cab 30:
http://www.thomann.de/fr/warwick_blue_cab_30.htm