I have to admit I don't know much about guitar. I started on guitar but never could progress far. I'd eventually put it down and forget about it for a while. When I really put some time into bass I found I just loved it and couldn't put it down. It is a cool project to save it from the garbage bin. It feels good to play an instrument you've improved and worked on. Definitely post some pics. I just rewired my P bass with the premium wiring kit listed on this page: http://www.stewmac.com/shop/Electron...or_P-Bass.html
It started life as a Squire. I've upgraded the neck with a Warmoth bubinga neck with ebony fretboard and put a Fender Vintage Reissue split pickup before upgrading the J pickup and electronics recently. I had to buy new knobs for the wiring kit. I didn't think the pots would affect the tone and clarity as much as they did. I had to do a bunch of hacking under the pick guard to get the pots to fit because of the style of my P which you probably won't have to deal with and it was a pain but was totally worth it.
As for strings I run Chromes, which are flatwound, on my P and D'addario EXL160 Medium Gauge nickel round wounds on my Schecter. The flats give a nice, deep, mellow tone which works really well with the vintage pickup to give some old school tone. The Schecter has active pickups and with those strings I can make amps growl and bark. I think it's a bass that a lot of metal guys would love. I'd stick to medium or light gauge strings. The heavier ones can take their toll on a neck.
As for the pickups you might dig the Fender Vintage reissue if you like a more classic tone. Fender has some hotter pickups or you might like the Seymour Duncan Quarter Pound or their hot pickup if you like a more aggressive or modern sound. They're decent sounding but aren't insanely overpriced which I find some pickups to be.
I'd go to a store with a bunch of basses and fiddle with some and see if you can find one you like before plunking down money on a pickup. They cost so much. I knew what I wanted and bought the Fender vintage without hearing it and it's everything I hoped it would be.
I look forward to seeing some pics of your progress!