Understand Your Mileage
Start with looking at what you will be expected to do. For example, on average everyone will run between 12 - 18 miles split over three legs (each leg is 4 - 6 miles in length on average). To that end, being prepared enough to run a 10 miler is more than ample to cover the distance. There are many half marathon training programs, most run about 8 – 12 weeks long. An example of one can be found on Runner's World. That said, there are certainly plenty of people who do just fine without running beyond 6 – 8 miles as their maximum distance in any single run.
Run Some Doubles
We would also encourage you to run 1 - 2 doubles each week for the last 3 – 4 weeks before the relay. For example, once you know what your legs will be (especially your longest leg) schedule two runs within an 8 – 12 hour period in which neither run is as long as your longest leg. Again, there are plenty of people who do just fine without running any doubles. If you have a good, solid base of miles, you will probably be just fine. However, if you want to know what it will really feel like, consider running a few doubles.
Pace Yourself
When it comes to pace, you should be able to hold your 10K race pace for your first two legs quite easily (assuming you do not have a lot of hills), especially if you have incorporated some doubles in your training plan that individually are NOT longer than your longest leg. For your third leg, you will likely not be able to hold your standard 10K race pace because you will be tired from the previous two legs and lack of sleep. For your third leg, go out a little slower than your normal 10K pace (perhaps 10 – 15 seconds per mile slower) for the first mile and assess how you feel at the end of that first mile before deciding to ratchet up the pace or back off just a little more.
Taper Before The Relay
Like any endurance event, make sure the last full week of training before the relay is a taper. Trust us, you will be thankful you rested. Ideally, your taper would begin two full weeks out or you might bonk!