Sometimes all you want is to get out of the house, but when the weather is nasty it's not easy. The Mawson Club is quiet, there are no lines to get in, the parking is right outside the door, and it's a nice temperature. The kids' play area is a decent size, has stuff for a wide range of kid ages, has a nice foam floor to fall over on, and (if you have any energy left) has a ball pit. There is no door, and you're not allowed to eat or drink inside - but there are chairs and tables just outside, so you can gulp something down with as much leisure as you get at home. It's a particularly good place to meet with friends, whether they also have kids or not.
Canberra's one lack is that there's no beach. But we still have beautiful parkland circling water views. Wherever you are in Canberra, there's a lake nearby - with BBQs and stroller-friendly paths. Park anywhere, bump your way across grass and/or tambark towards the water, and once you find a path walk in either direction until your baby is either bored or asleep. It's great for their Vitamin D, and for your health. It's also a great place to go with friends and/or for a picnic.
Okay, this one is expensive and you'll definitely need a stroller or a minion to get your midget from the carpark to the play area but it's worth it (maybe even worth a year membership, especially during Winter). The whole thing is fabulous (Awesome Earth may frighten littlies, so keep an eye on the timer outside the artificial lightning) but what you want is "Mini-Q".
There's an awesome playground (with mesh, so they can't fall out), a water-play area suitable for all ages (there are smocks, but take a change of clothes anyway, and a spare shirt for yourself), awesome make-believe areas built to tiny-person scale (including a bakery, pet shop, and construction site), and a sensory play area that is perfect for active little ones.