Welcome aboard
A P/J bass should sound exactly as P-ish as a P-bass of the same construction, if you set the J output to zero.
It is worth looking at second-hand basses. See what's for sale here, and use the Feedback section to form an opinion of the seller. Most people here are honest and sane, and you'll generally get an honest opinion of what the thing is like from a player's viewpoint as well.
In my opinion you can't really go wrong with an Ibanez SR300 as a starter. They are active, and you may think they look too metal-ish, but to my mind they have no other disadvantages. The neck is skinny, which may not be comfortable for you. (FWIW, I don't like metal...) Personally I want something which plays well and comfortably and sounds good, and I'm not too fussed what it looks like, but it depends what you want to do with it.
Also worth looking at are the Peavey Milestones. Again, skinny neck.
I'd try some basses out, to learn what you like as a player. Be aware that your taste may change as you learn. My experience suggests there is no bass which is "just right" forever; so it depends what you need.
Good luck, and do ask questions