Due to a finger joint problem, I need to use a pick most of the time. I like large triangular picks, which allow better control and stronger attack (if desired) because of the larger gripping area. I also don't drop them as often. My faves are Clayton 1.26mm. The thicker the pick, the thicker the tone IME. I use 1.00mm sometimes too.
Lately, I've been recording from the DI on my G-K 700RB-II. I use post-EQ, and adjust the "boost" control for a little more grit and thickness. When using a J, I'll sometimes boost the low mids, but never the bass, because it gets too boomy real fast. I usually cut the bass a little.
Experiment with hand position. Picking back by the bridge is great for lots of stuff, and necessary for palm muting. For fatter tone, move up toward the neck, often using a lighter touch. For good string-to-string balance, I'll sometimes pick diagonally, moving closer to the neck when hitting the D and G strings for more fatness there.
I have some rubber Wedgie picks, but never use them because of the dust someone else mentioned.
Edit: Don't forget a little compression! I like my Aphex Punch Factory, but I'm sure better ones are out there.