If you have only a "little bit of fear of heights" but have gone up some peaks, I think you can probably be OK on Half Dome. I would describe myself that way as well. My first trip up Half Dome I really didn't have a clue how steep the final part was till I got there and saw the little ants (or so it looked) going up the cables and realized how steep it looked. At that point my stomach dropped out and I thought "no &@(#&@#ing way!!) but I'd come all that way with my friend, so we started up the cables. I was wondering "how am I gonna do this" when I heard my friend behind me say in a quavery voice, "Debbie, I don't know if I can do this." Suddenly I got much braver, told her to just concentrate on her feet and the next step, and not to look down or way up ahead, and we both made it to the top just fine. She brought her boyfriend up later that summer, and I came up a couple more times after that.

My advice, other than to not look down below you or too far up ahead during the cable portion, would be to bring some of those lightweight gardening gloves with the rubber grip (either dots or solid, I've always used the dots but the solid rubber palms look like they'd be great). Also be prepared that when you are going up and others are coming down, you will only be able to hang on to one side, not both cables. And when you come down, you may want to go down backwards. I'd thought I'd have to do that, but found I was fine looking down (not too far ahead, again), and that it had been the climb up that was mostly the problem for me. Once I'd done that, I felt MUCH better.

I second the idea not to do the cable portion if it's raining or threatening a lightning storm.

I hope you will get to enjoy the thrill of standing atop Half Dome and feeling on top of the world. It's definitely worth the pre-cable nervousness!