A Precision bass is a great allrounder and an excellent choice for what you need.
The American Pro II Precision is a very good instrument and will serve you well, although you may find basses that are better VALUE at that price. You tend to pay a little extra for the Fender logo on the headstock.
I disagree with everyone saying you should get a cheaper bass. Cheap basses can be very good instruments but they often need a bit of work to get the best out of them: Fret levelling/crowning, setup and maybe even things like upgrading the pickups. The problem is that many beginners don't even realise this and may struggle for years on instruments that could have been made better after 1-2 hours with a luthier or a professional guitar tech.
There's also the fact that cheap basses can vary a lot in quality, even within the same model and a beginner won't be able to tell the good ones from the bad ones. Ironically, finding a really great, cheap "beginners" bass requires an experienced player.
If you have the money to buy a good quality instrument like a Fender Pro your learning will be easier, more inspiring and you will know that the quality of your instrument isn't holding you back.
I never understood the argument that beginners should start out on cheap basses in case they give up playing. Sure, a lot of beginners WILL give up playing bass but maybe they give up precisely because they are struggling with a poor instrument? If you can't afford an expensive instrument, sure, get whatever is within your budget but always get the best instrument you can afford. Resale value will generally be good for more expensive instruments (at least if you buy a classic model such a a Fender P) so even if you end up selling it you won't lose all your money.
Having said that, if I had the choice between cheap bass+expensive amp and expensive bass+cheap amp I would spend the most money on the amp. Cheap amps/cabs are often rubbish while cheap basses can often be very good, or at least made into something good after a professional setup/fret levelling, etc.