Just a bit of a warning: Tokyo is not very baby friendly (I'm a mother here in Tokyo). And August is very sticky and humid. I don't recommend going outdoors, plus it'll be summer holiday so most places will be extremely crowded. If you can I suggest you visit in September.

Maybe Suginamiku is a good suburb of Tokyo to stay in. They run on the sobu line which is one of the least congested lines and pretty much connect to all the big stops in downtown Tokyo. But that being said, most places are really narrow. It's hard to travel with a baby pram in the city unless you're used to it. I usually use a baby carrier.

Hakone! It's abut 2-3hours by train and it's a nice little get away.

P.s. try downloading Mama's map on google play, it will show you all the nursing/diaper changing rooms near you (all stations should have one INSIDE the ticket gate). This app is so useful!