From my very limited (but on a crash course learning quickly) garden and grass knowledge...... yes I would try to eradicate the weeds before planting grass, otherwise they will continue to grow up through the grass. But check out what you use to kill the weeds and make sure it doesn't leave the soil unstable for grass to grow. We returfed our backyard, but our mates bought runners, I think like thin strips of turf and planted them for them to grow out.

I'd leave the planting of flowers/vege etc until you live there so you can give them the attention they'll need, and maybe just focus on the clean up and grass growing for now? For the swings and tramp it could always be over a bed of soft fall or sand, doesn't have to be grass. Less to water then too!

Good luck - the possibilities are endless and it will be worth the end result