What do I do?
Okay, so you've found a guy who is a motorcycling god, you're totally safe with him, and you've got the gear. We're ready to go now, right? Riders, before you take off, have a talk about what to expect and how the passenger should behave. You can help your lady friends (and friendly ladies) a lot by sharing a few basics that will more likely make the experience good for everyone. I've learned this stuff over the years, but it would've been nice to avoid confusion and have a handle on it before getting on the bike, where you can't really ask questions.
Always get on motorcycles from the left side. Don't attempt to mount the bike until the rider has indicated he's balanced and ready for you to do so. Ask the rider about how he would like you to get on the bike, and do everything smoothly and slowly and you won't cause any sad, stopped motorcycle tip-overs. If the bike is really tall and the rider is aware you are going to do so, you can put your left foot on the passenger peg closest to you, then step up and lift your right leg over the seat to the other peg. While we're on the subject, dismount in the same manner. Slow, steady, and when the rider indicates you should.
Your hands and arms
The best place to hold on depends on the motorcycle. If the bike has grab rails, holding them in positions that allow you to lock your elbows makes you feel more stable. Often reaching around and bracing yourself against the gas tank on a sport bike is necessary as a passenger when slowing or stopping. Grab rails on sport bikes are good in theory, but they aren't very functional and you're probably going to have to hang on to the rider. Putting your arms all the way around and clasping them in front is the probably your best bet to feel most secure. Now isn't the time to get shy.
Your feet and legs
Put the balls of your feet on the passenger pegs, and keep them there at all times. Rather than just loosely sitting your feet there, it's usually more comfortable to have an engaged stance so you can stand up a little when going over bumps. When you're on the bike, keep your knees pulled in tight against the bike and/or rider to create stability. Never, ever try to put your feet down when the bike is stopped. Keeping your feet in place keeps everything stable and gives you a good foundation for balance. I never knew how important those pegs were until my inside passenger peg closed up mid-way through a 40mph corner; keeping my foot off the road and staying on the bike until we exited the corner and I could get the peg back down was no small feat.
Your body is a mass than affects the balance of the bike. Don't wiggle and jiggle all over the place, and whenever possible, keep your head directly behind the rider's. Pay attention to where the rider leans, and lean with them. Don't lean in the opposite direction because you're scared the bike is going to tip over. It's not going to and by leaning the wrong way you'll make it very dangerous for the rider because they're actually going to have to lean even further to compensate for you. The faster the bike is moving in a straight line, the less your movement will affect the balance. For example, it's much better to adjust your seating position while cruising down the freeway at a healthy pace than while stopped at a stop light, or in a corner. Speaking of stop lights, it's tricky to balance that sucker while it's stopped and a passenger is perched on top of it. It's okay to turn your head, but try not to turn your body. And if you want to talk to the rider, a tap on the shoulder is usually all it takes.
It can be hard to communicate while riding, especially with full face helmets. Arrange a few mutually understood hand signals for anything you're concerned you might need to tell the rider while in motion. Usually, if something needs urgently discussed it's best just to indicate for the rider to pull over, and then deal with it while you're stopped. To tell the rider to pull over, tap them on the shoulder then deliberately and clearly point at the curb. If the rider is going too fast and you get scared or they make you uncomfortable, a swift punch in the kidneys is always fun, but it's probably more effective (and nicer) to put your hand out palm down in front of them where they can see it and slowly push down a few times. They'll get the message. If you're a daredevil and you're ready for a real ride, palm up and pull your hand up quickly a couple times to indicate you're comfortable with more speed.
A note from Sean to riders about braking with a passenger:
When you're carrying a passenger, your rear brake suddenly becomes a lot more powerful. The weight distribution is shifted significantly up and back. Because of that, you should use it more, and by doing so you'll be able to brake nearly as fast as you would without a passenger on the bike.
Now I'm getting the hang of this
Once you have the basics down and you start riding pillion regularly, there are some great benefits and you can also have a whole ton of fun. Rather than being a passive bystander, get engaged with the ride. Look out ahead as much as possible. You can start to identify upcoming changes in the road surface and adjust your seating position as the rider does (rise up off the seat for bumps and dips). If you're interested in becoming a rider yourself, take advantage of this time on the bike to get used to the sensations and make note what the rider does in every situation. Riders, if you enjoy riding fast and canyons, you can start to teach your passenger about leaning off the bike with you in turns, leveraging the extra weight for some truly fun cornering. I would give you more advice about getting fast riding two-up, but it really comes down to a very specific poetry between whatever bike you're riding, and the body positions of the two riders, and what works for Sean and I is most likely not applicable across broad situations.
I'll end on this note: my casual gender assignments (men as riders / ladies as passengers) might make you want to call me out for sexism, but when I see guys riding two up, it's usually for practical purposes; someone crashed or bikes are being transported around. And, well, I'm a girl and I like to ride bikes with boys.