Adelaide has two kinds of day that drive you indoors, and the second one surprises visitors. Rain is the obvious one. The other is heat: summer here regularly passes 35 degrees, and an unshaded playground at midday in February is genuinely unusable. Both call for the same list.
The good news is that the free ones are clustered together on North Terrace, so a wet morning can be spent entirely under cover without moving the car.
Free and indoors on North Terrace
The South Australian Museum is free, open 10am to 5pm every day, and holds the things small children reliably stop for: the whale skeletons, the Egyptian room and the opalised fossils. It is a short walk from the Botanic Garden, whose Bicentennial Conservatory is glass-roofed and warm, which makes it a legitimate rainy-day option in its own right.
When they need to burn energy rather than look at things, these are the places built for it. Most charge per child, most have a cafe attached, and nearly all get busy on a wet weekend, so going early is worth it.
Sometimes you just need to sit down somewhere warm for an hour. These have a play area, a cubby or a fenced garden, which buys you a coffee you can finish.
What is there to do in Adelaide with kids when it rains?
The South Australian Museum and the Art Gallery of South Australia are both free and both on North Terrace, a short walk apart. Indoor play centres are scattered across the suburbs, and the Botanic Garden's Bicentennial Conservatory is under glass.
Is the South Australian Museum free?
General entry is free. Some special exhibitions are ticketed. It is open 10am to 5pm daily, except Christmas Day and Good Friday, and opens at noon on Anzac Day.
What do you do in Adelaide when it is too hot?
The same list as a rainy day. Adelaide summers pass 35 degrees often enough that unshaded playgrounds are unusable from late morning, so the free indoor museums and the play centres become the plan rather than the backup. Water play is the other option.