You could always do hardwood, which is pretty popular in my area. The newer vinyl/laminate 'hardwood' floors could also work

Cork on the correct underlayment is warm and sound deadening.

Cork is a great option for a basement....they are relatively soft underfoot, sound deadening, and naturally mildew resistant. They can dent easily, though, so that could be an issue with kids and/or dogs. Some brands of cork have more durable finishes than others, but they are all prone to dents/chips. That said, most dings "self heal" -- cork is pretty neat.

Other floating floors like vinyl tile/vinyl planks or laminate are popular choices for basements. Tile would also be great (the wood looking tiles are pretty nice, too), but I wouldn't want tile in a basement if they weren't heated, at least where I live. I wouldn't do wood.

My basement has hardwood, never noticed any noise coming upstairs. However oddly my basement is usually warmer then the rest of the house and im in canada so it gets very cold

Check out carpet tiles by FLOR.COM. They have some new designs that are really nice and this would be a perfect choice for your family. You can design any look you want, any colors, and if there is a "opps" you pull up the "tile" and clean it off and let it dry and replace it. If it is damaged by some permanent problem, you simply replace the tile. Just order several extras and no problem. Has a nice backing so would protect from cold. You can order samples on line and check it out.