I have not had the experience of three under three but I did have two under two plus a seven year old. Still, I think I can offer you the advice you seek. Here are some tips:
• Try to view your wife's day the same way you would if she worked outside the home. Caring for three young children and the home is a full time gig. If you can, allow her a reprieve when you come home. Even a half hour will most likely be appreciated. If you can handle more, great, but if not try to give her as much alone time as possible. After all, you might be tired from working your 8 hour shift, but remember: your shift is over; hers is not. Being a stay at home mom is like never leaving work; breaks are essential for survival.
• Take time with your newborn when s/he is not eating (if bf) or, take over feedings if bottle fed. This has a dual purpose: one is to allow you wife time with the older two and the other is so that you can bond with your baby. If you do all the feedings in the evening, you and your wife are getting equal time with the baby. If she is breast feeding, do everything else but the actual feeding. Change the diaper, burp, rock to sleep, and play when the brief but precious opportunity arises.
• While the baby is still waking up frequently at night, do your best to get up at least once during the night. Of course, you need to be sharp at work, but believe me she needs to be sharp to keep an eye on two young, mobile children. They are at an age where they are most often looking for trouble! Talk with your wife. See which feeding she'd most like to skip. My husband was awful about waking up in the middle of the night so I had him do the early evening feeding while I hit the sack at the same time as my two year old (7:00). We breast fed, so I wasn't completely off the hook, but it was no big deal to let the baby nurse while I snoozed while daddy waited to burp, swaddle, & then put the baby to bed. This allowed me a 5 hour block of sleep, which any new mother will tell you is like winning the lottery.
• In addition to bonding with the baby, you should try to spend one on one time with each of the other kids as well. It's tough for the kids to have a newborn dominating all of mommy's time. Both of your kids are at an age where they are prone to jealousy. It will make them act out during the day (and night too, but it's three against one during the day, so it's more unpleasant then) against your wife or the baby. Making sure everyone feels "equal" goes a long way towards reducing strife.
• try to prepare an extra meal or two on the weekends. It will give you both a night or two off from cooking from scratch. This frees up an hour or more which can be spent with your kids, catching up on housework, or whatever. It's so nice on the nights when everyone is tired to not have to cook, and reheating is MUCH easier and cheaper than hauling three kiddos out to a restaurant. Or, your wife can use these meals during the week at lunch time. How is this helping caring for the kids? It's not. It's taking a bit of the other load (the housework) off your wife, which makes caring for the kids a little easier.
• If you have relatives or close friends close by, ask them to come by during the day if possible. It's always nice to have adult company (for your wife) and if they have kids it's good too for your kids to have other kids to play with. This will help them blow off steam. I realize that this isn't some thing a father can do directly, but the underlying theme here is to arrange for help at home when you can't be there. I was always too stubborn to ask for help. Maybe your wife isn't, but in case she is...that's where you come in.
• baby proof the house. Your wife will appreciate fully operational baby gates, safety locks on the drawers and cupboards, and outlets with covers. It's one, or twenty, less things to worry about. It's also comforting to know the kids are safe. Happy, non-worried parents are better parents. (This is another one of those "indirect" help items.)
• If you need to run errands in the evenings after work, take along a kid or two. Neither my husband or I have run errands in over 4 years without at least one kid in tow (with a few rare exceptions). It is a nice way to accomplish some of that one on one time. (Either her home with one, or you out with one.) My middle child and my husband go grocery shopping every Saturday like clockwork. It's a chore, sure, but not as much because it's their special time together.
• My last piece of advice (which is very general, I know,) is always remember that you love your wife, and she loves you. Take her out to dinner, or whatever, without the kids. You two need to have a strong partnership now that you are outnumbered! Remembering your love for each other as individuals will help you both be better parents. Both for you and your wife, it may not always feel like you are doing enough, or the right thing (for instance when you've followed the bedtime routine exactly and they won't go to sleep) so you both need to support each other so you don't go crazy. All parents can use this advice, but the more kids you have, the more important and harder this becomes.
I hope that my advice helps you. I think it goes without saying that you and your wife have a few tough years ahead of you. Some days it'll feel like you are on the road to nowhere, and other days you'll wonder where the weeks went. Enjoy as much if it as you can. And take lots of pictures!