Tourist centre is usually pretty helpful. Are you taking / hiring a car? If so there are some beautiful peaceful places to stay outside Lerwick. Avoid the big hotels.
Shetland museum has lots of toddler friendly things to do and Clickimin leisure centre has toddler pool, flumes, outside pool, very small soft play area, cafe. Lots of playparks in Lerwick and every small village has one. Havly and Isleburgh cafes have kids play corners.
St Ninians beach is a must see but there are loads of other beaches you will probably have to yourself. Great for bird and whale watching, searching for crabs, fish, interesting shells.
Go to a Sunday Tea and if you are lucky they will be locals playing music - Vidlin ususally a good bet. Check here for whats going on locally.
This site has an accomodation bit which has the odd holiday let place.
Windproof clothes yes. Bring both warm and sun hats for the children as you never know what the weather will be like.
Go to one of the fish shops - don't buy the stuff they have on show, ask if they have x and they will probably get something super fresh out of the back for you.
Also the (pet) fish shop in Lerwick is a big hit with toddlers.