I have a 3.5 yr daughter and she was the main reason I built up my DSLR kit
For the low light indoor situation I bought Sigma 30 f/1.4, it's worked wonderfully. You can use large apertures (below f/2.8) until your child starts to walk since they can't move fast until then. It gets harder when they can walk and run away
I also purchased a Metz 48 flash with my camera. Even with a fast lens, bounced flash (off the ceiling or the walls) can make your pics just better - good light(ning) is a key. After a little practicing bounced flash pictures look quite natural (you find the balance between ambient and flash light). If you don't have a flash with swivel head already then I recommend to buy one. With bounced flash (or fill flash outdoors) you can get great photos even when using the kit lens.
Later when your child gets very mobile, a good standard zoom becomes handy to chase him/her. Originally I had a Pentax 17-70 f/4, combined with the flash it worked fine. I regret selling that lens. The 16/17-50 f/2.8 zooms from Tamron, Sigma and Pentax are also good. Without flash f/2.8 may not always cut it indoors (especially with older bodies having not that great high ISO), on the other hand you may need f/2.8-f/4 for good DOF (to hide smaller focus errors) when chasing the kid. Of course you can still use even f/1.4 if the child is not moving and busy doing something (which is quite rare with my daughter, she is very active and always runs away at first sight of my camera lol).
So I'd recommend to buy (in this order): a flash with swivel head if you don't have one already (you don't need top-of-the-line expensive one), a fast prime (Sigma 30 f/1.4, Pentax Fa 31 f/1.8 if you have lots of $$$ to spare, etc), later on a good standard f/2.8 zoom.