Choose and learn about your route
First, be sure to learn as much as possible about the route you want to ascend by talking to employees at your local climbing shop or guide service, perusing guidebooks or websites that offer detailed descriptions of routes in the area you want to climb and/or talking with local climbers who have done the route.
Choose a route that is well within and even easier than your single-pitch route abilities. For example, if you regularly climb one-pitch 5.9 trad routes, choose a 5.6 or 5.7 multipitch route to climb. Multipitch climbing is not as straightforward as cragging one-pitch routes because you can't always see where the route goes from the ground. Also, it sometimes feels as if the pitches are harder simply because you are higher off the ground and you have more details---such as making anchors, belaying from above and managing the rope---to deal with. Finally, start with a shorter multipitch route so that you don't get in over your head. For example, do a three-pitch route and figure out how to manage your rope or create a solid, multipoint anchor while at a hanging belay.
Make sure to draw out a detailed topo of the route you plan to do, including descriptions of both the ascent and the descent. Sometimes multipitch routes can be rappelled, but often you have to walk off, in which case you should really have a good idea where to descend.
Plan your day carefully
Check the weather and make sure conditions are good. More than likely if you haven't ever climbed a multipitch route, things are going to take a lot longer than you thought. Thus, if the weather is going to take a turn for the worse after noon, you should consider doing single-pitch routes.
Think about how much food and water you want to bring, and any extra layers of clothing you might need. You will inevitably bring too much or too little water, food and clothing, and so it is to your advantage to start small in order to figure out exactly what you need. Also think about the footwear you need. If the descent is long you will need sneakers, but if you are rappelling you can leave them behind. Finally, figure out how many ropes you will need. Some shorter routes only require one rope (typically never shorter than 200 feet), and you can rappel off any pitch with one rope. However, other long routes require that you have two ropes for descending.
Get to the crag as early as possible
Next, head out to your chosen crag as early as possible. If the weather is good, get to the parking lot at daybreak. This is a good idea because it will give you plenty of time to do your chosen route, and it will ensure that you are the first person on the route. There's nothing worse than making copious plans for your first multipitch route, only to get there and find there are already three parties ahead of you. Easy routes in popular areas tend to see a lot of traffic. This can also be dangerous as you are more likely to be subjected to rock (or gear) falling. Don't forget to bring your helmet.
Double check everything before you do the route
When you get to the base or your route, double check everything with your partner (make a written list if you're the type of person who always forgets things). Remember, if you forgot your fleece jacket, for example, you could be forced to descend before finishing the route if it gets too cold for you to continue. Often the temperatures at the base of a cliff are significantly different from the temps at the top of a cliff. Also, forgetting your water can lead to dehydration, which can actually be dangerous.
After you have double checked everything twice, taken one last bathroom break (you won't be on flat ground again for a while), hung your extra food on a tree to keep it away from animals, and coiled your ropes neatly at the base of the wall, get started. While climbing, be sure to focus on your systems. When you reach the anchor, make sure to coil your ropes as neatly as possible. You can do this by making a small, compact pile on a ledge or by coiling the rope over something (some people coil ropes of the daisy chain or piece of webbing that connects them to the anchor point). The last thing you want to have to deal with is a bunch of messy ropes all knotted together. This is especially important if you have two ropes (bring ropes of different colors).
Things to consider while climbing
As you're climbing watch the weather and take note of how your partner is feeling. If he is getting really tired or spaced out, assess the situation. Should you keep going? It could be dangerous if you do. Or, are scary, lightning-filled clouds racing across the sky toward you? If so and you are moving slowly, you might want to consider rappelling down immediately. Finally, if possible, try to climb your first multipitch route with an experienced climber. There is much to learn that cannot be covered in a short guide. It's always helpful to have an expert nearby from whom you can ask loads of questions.