For the record, I'm not saying this is the absolutely correct way to put on tape, and I know other people who do it a bit differently. But this way works for me and goes by the generally accepted standard way of putting on bar tape, and it will stay in place until you're ready to change it out again.
1. GET EVERYTHING LAID OUT Make sure you have your tape, sissors, and electrical tape laid out and ready to easily access. This also includes pre-cutting tape that you know you'll need (it's a headache to hold the handle bar tape in place).
2. TAKE OFF OLD HANDLEBAR TAPE Pull back the hoods. Snip the old electrical tape careful not to scratch the handlebar paint, and then pull off the old tape. Throw it away so it's not in your way (or if you're lazy just toss it in a pile).
3. UNWRAP NEW HANDLEBAR TAPE And go ahead and start peeling off the adhesise strip on the back.
4. START AT THE BOTTOM OF THE HANDLEBARS Using a piece of electrical tape, attach one end of the handlebar tape so that it lines up exactly with the edge of the handlebar, and then wrap it towards the outside for one full turn. Then continue to wrap it at an angle up the bar - the sticky adhesive on the bottom of the tape should always be in contact with the handlebar surface.
5. SHIFTERS When you get close to the shifters, place the short pieces of handlebar tape underneath, and then continue to wrap around. When you get to the point that the tape would cover the short pieces of tape, go further up around the tape and then continue along the top of the handlebars.
6. FINISHING When you get to the point that you're covered as much of the top of the handlebars as you'd like, hold the handlebar tape taut forward at the same angle you've been wrapping, and cut it at a 90 angle to the handlebars (so it will be like a triangle shape). Then attach a piece of tape along that cut edge, and continue wrapping it so that you have a continuous edge. Overlap the electrical tape enough that it will both hold the handlebar tape in place, while securely attaching it to the handlebars.
7. FINAL TOUCHES Snip the short pieces of handlebar tape so they don't cover the holes that secure the hoods in place, and pull down the hoods. Throw all your trash away. And enjoy how clean and sleek your bike looks!