Having returned from 10 days at Aitutaki here is my families advice.
Transport - We walked a lot because we had a house close to shops. This had several advantages, we got to talk to locals, we went into shops we would unlikely to have stopped at otherwise and we found some really good local eateries. Also the locals do not walk so as a consequence I found $10 in change on the road side. Rental cars can be in short supply which is why many people hire the bikes. The local speed limit is 40kmh but even so many tourists fall off their bikes especially when they go inland which are all coral roads, so be careful. Otherwise hire a bicycle the island is small and it is a pleasant way to get around.
Food - We ate out a lot at local eateries and had no issues at all. We have four kids and I know many people get nervous about eating out because they do not want sick kids, but I recommend going to the various local cafes. As we had our own kitchen we cooked a little so purchased local produce, again all was good quality.
Water - Again know many people get nervous because they do not want to get sick but we drank the local water and had not problems,
Mosquitos - Yes there are plenty and we all got bitten at some stage but have no issues. Bushman's with DEET is the best protection.
Money - Take cash. The card machines were out of service for a number of days when we were there so we had to go into town to get more cash. Beware that this can occur so take extra cash. Also a lot of places, including the lagoon tour operators only accept cash.
Shopping - We found souvenirs, clothes, pearls at the same price or cheaper than in Rarotonga. The range is less and you have to look for them but they are there.
Activities - There are plenty. Not all are well advertised so you need to find them. Go see and island cultural show one night. Go explore and enjoy the experience. The locals are friendly and happy to help so do not be shy to ask for advice.
Weekends - Very quiet as many things are closed so plan ahead and make sure you have cash to pay for it. Good time to ride your bike and do a lap of the island stopping for a few swims along the way.
Lagoon cruises - Thing about what you want from the cruise and then book accordingly. If you want a general look around then the large operators will probably work for you. If there are specific thinks you want to achieve then book a private charter. As a couple the charter will be 50% more but if you are a group the prices will be very similar.
Enjoy your time on Aitutaki it will be a wonderful experience. It is far better than Rarotonga.