Start by removing the tape from the old bars. It is possible to save it, but you're better off buying a couple of new rolls. Don't forget to untape the cables too. Then remove the levers. There's a little groove on the inside of Shimano levers, lift the rubber hood and push an allen key in there (5mm I think). Just loosen it enough to slide it off the end. I'm not sure how to get Campys off. Then remove the bars from the stem.
To put the new ones on, simply do all the above in reverse (mounts bars on stem, mount levers on bars, tape bars). You'll need to mess about with the positioning of the levers and the angle of the bars (unless you were clever and measured the old ones). Once you're happy, tape the cables to the bars with electrical tape. Leave the lever bolts loose enough for fine adjustments later. then the fun bit: bar tape. The instructions with my last roll were pretty good, but I'll give you a few basic pointers anyway. This is for the left bar, reverse for the right. You'll need to fold the rubber hood up to get it out of the way, then cut a short piece of tape (about 4 inches should do) to cover the clamp. Start at the end of the drops, with the end of the tape on top, pointing right (the roll should be in your left hand, to the left of the bars). Wrap a bit more than once round, going counter-clockwise, hanging over the end by half the tape's width. Then start working your way up, overlapping by about a third of the tape's width. Getting it right around the levers and bends can be tricky, but don't worry if it's not perfect. Eventually you'll get to the bulge in the centre of the bars, at which point you should stop. You'll want to cut the end at an angle, so it sits square to the bar, with the end on the bottom (hopefully it will be pointing backwards, but no biggie if it isn't). Finally, wrap some electrical tape round a few times to secure the tape, and having checked their position, tighten the levers (it's amazing how different they feel without tape, so you may want to move them slightly).
Hope that helped.