I personally have always stayed centrally, generally on or close by to Sukhumvit road.  Do not have kids so we stay at hotels as opposed to serviced apartments - which in Thailand are actually about the same as hotels.  My favourite place to stay is Holiday Inn Phloen Chit, next door to the Intercontinental.  It is next to Chidlom sky train, and from there you can also walk across the to Centralworld (Sky walk) (biggest mall in Bangkok with lots of kids things to do in the mall ice skating etc.).  Sky walk also goes all the way down to the aquariam and MBK.  Also Pratnum markets are a 10 min stroll, these are really old local markets where you buy t-shirts flip flops and all sorts, great for a look even if you dont want knock off's. There are night markets setup all around the are are of the hotel during the evening, and then you can escape back into the quiet of your hotel when you have had enough.

Highly recommend the Sky bar based at the hotel in central-world, super amazing views, best viewed at dusk.

BTW July is a great time to go, considered the "rainy season" many tourists are put off by that.  Sure it rains hard but not like all day in NZ, its not uncommon to get a downpour for 25mins and then beautiful sunshine the rest of the time.

PS making use of skywalks is important as the footpaths are much higher in Thailand for the downpours, and often not friendly with prams in some areas.