I go to Singapore every few months and I am always on a budget. I stay in hostels most times, in dorms, which is a good cheap option as long as you are ok with that for the kids.
This will free up a bit of cash for outings, which after all, is the main reason you are going right?
Some places I like to go that are either free or cheap:
Japanese Gardens - beautiful landscaped gardens. Take a picnic. There is also a turtle museum inside which costs a few dollars. Free entry and right on the MRT line.
Arab Street/Kampong Glam - a great place to people watch, drink mint or apple tea. Its a pedestrianised area so kids will be OK. Cheap trinkets to buy here. Visit the Malay cultural centre whilst here. Short walk from MRT.
Macritchie Resevoir - another great place to go for fresh air. See monkeys here. Free entry.
The Southern Ridges - a walk through some of the prettiest and most interesting parts of Singapore. You may need to do the whole thing over a day or two if the kids get tired. Free.
Singapore Botanical Gardens - great place for kids to run around, feed the swans, see squirrels, waterfalls, lakes and escape the city. Free.
Boat Quay/Clark Quay - follow the waterways or take a ride in a boat and see the opera house, Merlion, and a host of other landmarks. Free and plenty of places to eat and drink.
Fort Canning Park - Walk around the gardens with some great views of Singapore, and visit the Battle Box to learn about the surrender to the Japanese in WWII - very interesting place. Fort Canning park is free entry to walk around. Battle Box charges an entry fee.
Little India - Great place to walk around and visit the markets and sample cheap indian cuisine.
Chinatown - the kids will love the colourful streets here.
Temples - temples are everywhere, free to enter and great places to experience local culture.
HortPark - a great place for the kids to see different types of gardens with butterflies and all sorts of activities. free water spouts for drinking (as there are in most of the outdoor places in Singapore, so take a bottle to top up).
Hope that gives you some food for thought!