Bear in mind that revenue canada takes a dim view of finishing rooms and claiming them. You can claim the costs of the room but should you ever sell your house those expenses will be considered capital gains since they were not part of your principle residence. Better to just finish the room and suck up the cost but then claim anything you put into the room that would be replaced such as carpet or shelving, paint for the walls that sort of thing.

Always cheaper if you do it yourself and consider making it child friendly - maybe not the prettiest but certainly usable. Cover walls with cheap board and paint the wood - no need to finish drywall. Use a cheaper carpet but upgrade the underpad for cushioning for instance or use laminates or vinyl flooring not hardwoods. You can get away with an unfinished ceiling if you want to.

To start I would do a small area at the bottom of the stairs and then gradually as time and money allow expand the area. To start still use the kitchen upstairs, etc. and again that will keep costs down. Kids don't need fancy and don't need as much space as some people think so it is doable.