It is sometimes nice to get a little wider indoors but unless you want to head into landscaping, I don't think the 10-20mm f/3.5 is the right lens. If you want a different 'standard' lens, Canon make an EF-S 15-85 mm f/3.5-5.6 IS USM lens that adds a little versatility at both ends but the gains may not be worth the expense.

I think you could probably be better off looking at a longer zoom lens for when the kids are playing sport or running around in the back yard. A good choice would be the Canon 55-250mm f/4-5.6 IS zoom. These are fairly cheap as they often come as part of a 2-lens kit with new cameras. Easy to pick up second hand or you should be able to buy a cheap new one from a debundled kit (i.e. where the kit has been broken apart and the lenses sold separately).

A flash is a good idea. If you don't want the expense of the Canon flash, I can recommend the Nissin Di622 Mk II. This costs around $299 in Australia or half that from overseas. The Canon version will work as a remote flash with Canon cameras that support that functionality. It also has an optical slave mode so you can trigger it with your camera's built in flash to provide side ligthing. Ideal for those family portraits.