I use dirt-cheap plastic tape on my aero bars. Wrap direction will depend strongly on the style of bar and your hand position. Some people really grapple with their aero bars and yank up on them when cranking on the pedals. Other people (myself included) rarely have anything more than a light grip on them to control the bike.
On my current tri/tt bike, I've got flat, "race bend" bars. I wrapped from the ends in (i.e. fore-aft), with the tape going down on the inside. I used Ambrosio tape I got from probikekit for like $2.39. Can't be beat.
The bigger question, IMHO: Where the heck do you put the little colorful trim-strip tape?
The way I see it, there are 4 viable choices: front ends of the base bars, back ends of the base bars, front ends of the aero bars, or back ends of the aero bars. Since the brakes/shifters are trimming off the front ends of the bars nicely, that just leaves the two inboard positions. Since I've got a computer mounted right where the tape ends on my left aero bar, that left just the inner ends of the base bars - so that's where I put the trim. I left the outer ends of the wrap untrimmed, since the tension of the tape holds it in place. The inner ends of the aero bars (and the base bars under the trim tape) are trimmed with black vinyl (electrician's) tape, which works well, since the bars are all black.