Woolwich and Plumstead are not pretty places to live. They're scruffy in many areas, down at heel and Woolwich particularly can be a bit untidy and unsightly due to large areas of social housing maintained only adequately rather than brilliantly by the council. Shopping facilities in Woolwich are generally good and getting better, but there are no posh delis, 'naice' cafes, trendy bars in the town centre, and only a couple of bars down on the Royal Arsenal development (which is very nice architecturally but also very pricey to live on.) Your friends will probably raise eyeborws when you tell them where you live; people will feel compelled to tell you stories about their cousin's boyfriend's brother-in-law who lives in the area being caught in gun crossfire on a weekly basis.
But I can honestly say that in 15 years of living in London, the 4 years I spent in Woolwich/Plumstead (Glyndon Estate) in a mid-level ex-LA tower-block flat were some of the best. I never felt unsafe, threatened or at risk anywhere - and I was a single woman living alone and used to get home either late or after dark much of the time, trundling my way through the back lanes of the estate and across the grassy area out front to my block. Generally speaking, the horror stories come from people who've never visited SE18 and who quake in their mataphorical boots at the idea of living anywhere but Fulham or Muswell Hill. That might sound scathing, but so much of the time their views were just plain ol' prejudice towards (shock, horror!) people with Saarf London accents, council housing, socially mixed neighbourhoods and anywhere without a Waitrose.
My estate was no noisier than anywhere else I've lived in London (apart from traffic noise and noise from City Airport - if you like pin-drop quiet, the airport will really put you off the area!) I generally found the atmosphere throughout SE18 quite pleasant: vastly mixed demographic of families, young professionals drawn in by lower rents than surrounding areas and older people who'd lived in the area for decades. I had a big grassy area in front of my block which used to attract neighbourhood kids for football, cricket and generally playing out. I never experienced any bother or antisocial behaviour in my time there.
For what it's worth, though you say schools aren't a factor for you, all the local primary schools are rated 'Good' by Ofsted. Pretty impressive considering some political wards in SE18 are among the 5% most deprived in the UK.
Property wise, the parts of Plumstead around the Common and up towards Shooters Hill are considered the most desirable in the area, particularly for young families. Property generally is excellent value - you can get a three-bed Victorian terrace needing cosmetic work for around the £120,000 mark and flat in some of the newer off-Arsenal developments from £100,000 upwards. And if you ever wanted to move on and let your flat/house out, the yields are enormous: I'm currently letting my one-bed flat, with a sale value of about £90,000, for £750 per month to a professional working tenant.
Transport links are great - the DLR from Woolwich Arsenal can get you to Bank in 25 minutes and Canary Wharf in 15, plus links you in to the Jubilee line at Canning Town. You can get from London Bridge to Plumstead by rail in 15 minutes on a fast train and in 30 minutes on a regular train. DP worked at Tower Bridge at the time and used to love going out in the area after work and then getting the riverboat back to my place in Woolwich at the end of the night!
I think whether or not SE18 is a go-er for you will depend on whether you can align all of the above.