I live in Bromley and to be honest, it doesn't feel very semi-rural to me, although you can be out in the countryside within a 20 minute/half an hour drive. Chislehurst is a bit more villagey in feel maybe but still quite suburban. There are some decent green spaces in the Borough itself.
Near me (close to Bromley North/Sundridge Park stations) there are quite a number of good and outstanding state primaries and also good private schools too.
State-wise, Parish, Valley and Bickley (a bit further from me) are all outstanding and St Joseph's (Roman Catholic) and Scotts Park are both good too. SP had a slightly poor Ofsted recently but the issues were mainly to do with management I think and as the school has had to expand from two to three form entry in recent years (as has Parish) this may have been part of the problem (I don't know though). A lot of people now recommend St George's (also Bickley way too) which has improved massively in the last few years.
Breaside is an OFSTED outstanding co-ed junior school and Ashgrove is very good too (I think it has quite a focus on getting children into the Grammar schools and super-selectives in the area). Other local preps include Bickley Park prep (all boys I think).
For girls, Bromley High (a GDST School) is known to be excellent and has both juniors and seniors. Farringdon is also good and I think goes all the way to 18 too, although I am afraid I don't know so much about it.
State seniors include Bullers Wood for girls and Ravensbourne and Ravenswood for co-ed. There are others as well but I haven't reached senior level yet so have less idea. I do know that a lot of people try to get their children into the grammar schools in Bexley and Kent but unless you choose to live in those areas your child is obviously going to have to travel to ge there (and I think will have to get better results in the 11-plus in some cases).
The super selectives in the area are St Olaves for boys and Newstead Woods for girls. Your child will need ot be seriously academic to get in and more to the point, to thrive, at either school, certainly from what I have heard.
I have only mentioned a handful of schools here but, to be honest, Bromley schools have a very good reputation generally. However, like most London boroughs, for state schools, there is a big issue with the number of available places.
I should mention actually there is a bi-lingual French/English free school opening in September - La Fontaine Academy - which might be of interest to you.
Travel-wise, getting to Canary Warf isn't too hard from Bromley. Trains either from Elmstead Woods/Chislehurst to London Bridge and then the Jubilee Line (or you might be able to change at Lewisham to the DLR) or if you are Sundridge Park/Bromley North way, you get the shuttle train to Grove Park and change there for London Bridge.
Bromley isn't the most atmospheric town you could end up in but it does have plenty of amenities and is convenient for a lot of places and has good transport links. If you like an eclectic area, Crystal Place is great at the moment or if you want to go more villagey, Chislehurst or Petts Wood (which is also on the same line to London Bridge) might be preferable.