the isle of dogs is an odd mix. some mega expensive 80s onwards stuff (canary wharf is on the island) and some of the old residents still left in the bits of social housing that haven't been swept away for yuppie riverside developments.
i'd say that south of the river (greenwich and blackheath are not that far away) may be more fun, although both greenwich and blackheath have got increasingly touristy / expensive. there are a few bits of greenwich park where dogs aren't allowed (more
here) but the rest is ok. and may offer squirrels to chase.
chrisp street market in poplar (north of the island) is more old school market. there's a bloody big asda near crossharbour DLR station on the island.
dogs travel free on TFL transport modes including buses and the DLR, but if you go on the underground, you are expected to carry them up / down escalators, as the bit at the end of escalators can be dangerous to dogs' paws (the signs saying 'dogs must be carried on escalators does not actually mean it's compulsory to carry a dog, but see
this video from 1 april a few years back)
it always used to be the rule that you had to go upstairs on a bus if you had a dog with you. i'm not sure if this is still the rule.
and dogs are not supposed to sit on seats on buses / trains etc either.
you can't pay a cash fare to the driver on london buses - you need either a contactless card or get an oyster card (you can buy these at a lot of newsagents in london)
dockland light railway (DLR) has self service ticket machines at stations - again, you can't buy a ticket on the train.
the cable car thing costs a bit more than buses etc, as do river boats. the woolwich ferry is free but may be a bit distant.
presume you'll discuss transport options with doggie's person before trying it - some are fine on buses / the underground, others can find it all a bit too much for them.
if you're visiting after 17 may, i'd recommend a visit to the
museum of london, docklands (has a nice cafe as well) although don't think you can take doggy with you.