There are several places in close proximity to Canary Wharf that could be of interest.
In London proper you have Blackheath and Greenwich areas, depending where you would be, you would have easy access by DLR (light train) to CW.
I actually have a 2 bedroom flat in North Greenwich, I lived there for a couple of years, and really loved it (North Greenwich is next to Canary Wharf in the Jubilee line, so commuting is relatively easy and would be a bonus if you need to run home errands of any kind).
Development in the Greenwich Peninsula is continuing now that the recession has eased a bit, so it is worth paying attention to this.
I know there are bigger flats for rent, there are many families with small children in the area. I don't know what the situation with schools is, so that could be an issue (local people send their children to the primary school in the peninsula, a big secondary school with an arts vocational component is planned for later this year).
The postcodes you mention are of areas that are in one of the poorest boroughs in London (and the country), Tower Hamlets. I would avoid E2 and E3 (around the notorious Hackney area, from where traditionally the Cockney criminal gangs come, nowadays it is simply run down and poor). Some places in E1 could be OK, but the problem there is that amenities are lacking.
E14 is in what used to be a very poor area, but the money from Canary Wharf is slowly rubbing in, so in close proximity to Canary Wharf and in the areas with river views you will have very good housing, but in other areas in the Isle of Dogs (that is the name of the area) you will find some of the problems associated with inner city living (some crime and some degree of gang related violence, social deprivation). Keep in mind that this is not remotely close to how bad it can be in some US innercities. Most of London is eminently liveable and enjoyable, I think E14 would be fine in general but open your eyes to make sure you spot any potential problem areas.
E14 has the advantage of having very good amenities (all the shopping possibilities in Canary Wharf, as well as big supermarkets and lots of small shops) easy access to Greenwich (which means museums and green areas), you have even got a farm there
London city farm Mudchute Park and Farm . Cinemas are easily available in Canary Wharf's outskirts (West India Quay), and Greenwich, including an art house, one short Tube ride away you have Surrey Quays shopping centre with more shops and cinemas.
I saw very smart houses in Rotherhithe (Canada Water tube), which has a shuttle ferry to Canary Wharf.
Some of my former colleagues with small children preferred life out of London, they were in Tunbridge Wells, South End or even Hove (close to Brighton) The commuting can be quite tiring (train+tube), and tremendously expensive, but many people are up for it.
You may have noticed I have not talked about car. If you live in London proper you may not need one (we have one and we use it only on weekends if at all, but we have no children).
If you live out of London you will need a car, which is quite expensive (as summarized by another poster).
Good luck and welcome to the UK.