We take a 1 week trip to Bali every year, with 2 little ones. We love it. Here would be my comments:
Ubud: Probably the most interesting part of Bali. Problem is, in terms of explorattion, it is one of the more difficult areas to go. Even the best parts of Bali are not particularly well catered to pedestrians but with a stroller it can be impossible. Ubud is in that category.
Kuta: Avoid at all costs. It is everything that Bali isn't, especialy at night. A sea of drunken Australia "bogans", (google it)
Seminyak: A great spot if you like dining and nightlife. Great restaurants and bars. Would particularly recommend Mama-San, Potato Head, (bar/beach club), and Bar Mexicola. The quality of the beach is only OK
Sanur: One of the nicest beaches in Bali, (white sand, good swimming, etc). And with the new traffic connection from the airport, a lot more convenient than it used to be. But parts of it lack charm. It doesn't have the nice aesthetics of Ubud or Seminyak.
Nusa-Dua: This is where we stayed when our children were very small, as we knew, for the most part that we would be confined to a resort for a lot of the time. The area itself doesnt have a lot of character or places to visit, but this tends to be where the resorts are larger, more pools etc. But I would not reccommend if yuo really want to see the highlights of Bali itself.
We now almost exclusively stay in Seminyak as it provides a good balance between adult and child considerations. But on our most recent trip, we tried Sanur and had a good time. However we found ourselves going into Seminyak at night quite often.
Couple of tips:
Organise, (prob through your hotel) the immigration "fast track" if they have not opened the new airport by the time you take your trip, (current airport is a turd)
Dont go out at night in Kuta
Do a bike ride from Mount Batur to Ubud, (might be able to put kids in a seat)
Visit a couple of art galleries in Ubud
Eat eat eat!
Take your wife for a sunset drink at the rock bar. She'll be gaga. Get there relatively early though, (1 hour before sunset) otherwise yuo may not get in
Hope this helps. Enjoy!