I live in the Upper E. Coast area and have a 21-month-old. The condo
complex has some playgrounds and pools, which are helpful; but he often prefers climbing steps and pushing buttons on the door keypads. (Fortunately there are lots of stairs and doors). There are baby/toddler music classes at the condo once a week and they're priced well at $12 per class but I'm not sure how long the instructor (a foreigner) will be around so that could always change. However, there are other music classes in the area that want $30-45 per 45-min session, which is crazy, esp since kids that age don't pay attention to a lot of the lesson anyway.
As far as free activities go, I've found the kids play area next to the restaurant at Ikea-Tampines to be great. They can play with pots and pans in a pretend kitchen, along with lots of other toys. Don't confuse it with Ikea's "Small Land," which is for older kids and located at the main entrance of the store. Another option is Parkway Parade shopping center, which has lots of toy ride-on vehicles on the second floor near all the shops for babies and kids. There is also an arcade two levels up. It's great to let them climb on the vehicles (as long as another kid isn't in there) and arcade machines. Blinking lights are entertaining. At this age, they don't know to ask you to put in a dollar so they can ride or play. East Coast Park is also nice for walking and to get your kid tired out, but be careful where you sit because there are lots of ants on the ground and on the benches. Public library is also an option.
I've found some activities on meetup.com but some of them are expensive but not all are. Many indoor playgrounds are also expensive but there are some good deals out there. I've found weekday mornings to be best for little ones at indoor playgrounds because the big kids are in school then.
Child care centers are very pricey. For a babysitter, if you don't have a live-in maid, there are some services but I haven't tried any of them. We usually get a friend or husband's coworker who likes kids to volunteer and babysit. You can't hire someone else's maid to "babysit on the side" because that's illegal.
Another thing, are you getting a car? If so, transporting your kids might be easier, although parking is usually a hassle. If you're going to use cabs, buses and the MRT, you'll have a challenging time trying to get your kids home from an activity in time for a nap or a meal. Public transport is good but moves very slowly compared to cars. However, you could sometimes also be waiting in "Disney World-style" lines/queues for cabs, esp if it rains. No fun when you've got a screaming kid and you need to get home immediately. You can do your best to time things according to schedule, but it doesn't usually work because EVERYTHING seems to be a huge production in Singapore. Because of all this, before we got a car, it was easier to just stay at the condo and not go anywhere.
Best of luck with your move and congratulations on your new baby. Hopefully you'll enjoy Singapore and all its quirks.