This is key to your sanity and any hope of getting some sleep. I know it may seem mean, but if one baby wakes up, wake the other one up and feed them together to keep them synchronized.
Sleep when the babies sleep.
Everyone says this, and it is very true, although much easier said that done. After you finally get the babies both to sleep, you will look around and see everything that needs to be done (dishes, laundry, birth announcements, thank you notes, etc.) or be tempted to take a few minutes to catch up on email or flip through a magazine, but fight that urge and go to sleep!!! Remember, this is a marathon, not a sprint, and you will need your rest. Everything else can wait. Really, it can.
Divide and conquer.
If you are anything like my husband and me, when you bring home your babies, you will be so in love with them that you won't want to miss a single yawn or coo. We initially both wanted to be there for every moment. While great in theory, we soon realized that this was a recipe for severe sleep deprivation (and crankiness) for the whole family. Certainly enjoy your time as a family, but if one parent can hold down the fort, send the other off for some much needed rest.
Keep a log. You'll be amazed how quickly you will forget which baby you just changed or fed when you are sleep deprived, and having a log makes it much easier to keep track of everything and to pass off care to the other parent or another caregiver. Initially we had a log with a column for each twin and a row for each hour. We would write down on the appropriate line when they had a wet or poopy diaper (important initially to make sure that they were getting enough to eat, especially since I was breastfeeding and it was hard to tell how much they were getting), which side they ate on (so I could alternate which baby got which breast in case one was a more vigorous eater), and how much they drank if they were getting any by bottle. The log got considerably simpler as we went along, but we continued to use some form of log for the first few months. This was especially helpful when I went back to work since I could come home and have a snapshot of the day.
The 5 S's (Swaddle, Side/Stomach, Shush, Swing, Suck). This is Harvey Karp's method for soothing babies (detailed in The Happiest Baby on the Block) and was my mantra for the first few months. It really does work and provides an easy to remember series of steps to follow when you are exhausted and dealing with a fussy baby.
Accept all offers of help.
This was hard for me since I like to be self-sufficient, but the generous help that we received from friends and family was invaluable during those first few months. I found the most helpful things to be those that allowed me to focus on the babies and/or get some rest (bringing meals, having our older kids over for a playdate, making a trip to the grocery store, holding the babies while I took a nap or shower).
Prepare yourself for the curious public.
Going out with twins in public (especially both sets) has made me empathize with celebrities. Despite the fact that twins are actually quite common these days, people seem to be endlessly fascinated with them, and the younger the twins are, the more fascinated people are. If everyone is in a good mood and you are not in a hurry, it can be fun to have people "Oooh" and "Aaah" over your babies and amusing to entertain their questions, but there will be times when you just want to do your grocery shopping and go home. In these cases, I have found that a pleasant aloofness works best. I just smile and move along, trying to engage as little as possible.