*warning I am typing on an iPhone
I can't speak much about woolwich but I LOVE Greenwich! It's like a little town but in one of the largest cities in the world. I would stick with west greenwich like cutty sark and by the Greenwich rail station but if you're looking for more space then westcombe park, maze hill and charlton (basically anywhere slightly east) are quiet but less expensive areas. Ride the 53 through east greenwich and you will small english town London that you think should belong in the country side. I always loved riding through there.
If you live on a hill you will have a fantastic view of canary wharf! During my first year in london, to relax I would take a stroll along the thames and gaze at canary wharf so I have a lot of good memories there. Greenwich park on a warm day cant be beat! Greenwich market on saturdays is fantastic to stroll through and on a cool day someone will be selling mulled wine which if you havent tried you must! Around cutty sark there are lots of small boutiques and souvenir shops. It is very different from central london but greenwich has a lot to offer. If you can afford north greenwich you will have access to the jubilee line which is more reliable than the south eastern line. North greenwich is a little trendy but mostly because everything is newly built. And you will get to enjoy the O2 center which has a great cinema, loads of restaurants and various little shops and bar/club. Bands play there often so it's handy to be able to get there easily.
The only bad thing about Greenwich is the new cross and deptford area to the further west. They are not the worst areas in the world but as a young woman I would rather not live there again and I am tuff broad
those areas are avoidable but worth mentioning.
If you have any questions feel free to pm me
good luck with the move!