I moved away from Alverstoke last year, I lived and worked there and in other areas in Gosport for years.
Do you know which road/area you are moving to? PM me if you like. Alverstoke is lovely, some areas of Gosport are grotty (like most towns) and depending which road you'll be in can influence schooling and GP registration.
Schools are ok, some are better than others but many are oversubscribed so getting a place will be easier said than done. Alverstoke Infants and Juniors are highly regarded, Gomer is good.
Also are there any areas around the Alverstoke region that should be avoided? Not really, some of the creek area that is classed as Alverstoke isn't as nice as other bits but none of it is too bad.
Medical and dental care? Dental care can be hard to register with if you need NHS dentists, they opened a new one in town which is good and was taking new patients last time I looked. Doctors surgeries are OK, I recommend Stoke Road or Brune Medical Centre if you are in catchment.
Activities for children? Sport opportunities? Lots going on, depending on the ages of your children. Sailing club, sports clubs, holiday clubs.
Tuition (music, languages?) No idea sorry.
What is the community like? I loved it there, I had great neighbours and no problems. Lots of Alverstoke has plenty of elderly wealthy people, but also young families and children. There is cubs, brownies etc as well as toddler groups and evening classes.
Shopping convenience? Gosport town centre is rubbish but they have a Superdrug, New Look, Clarks etc. Morrisons, Tesco, Lidl, Aldi and Waitrose (a small one) and a huge Tesco, Sainsburys and Asda in Fareham so not far. Easy enough to get the ferry over to Gunwharf Quays for the outlet, and if you drive then Fareham, Portsmouth, Southampton, Chichester and Winchester aren't too far.
Commuting to London? You'd have to get the ferry over to Portsmouth Harbour (The Hard) and then train up to Waterloo. Driving wouldn't be much fun but is do-able. Traffic on the peninsula is hideous and I don't miss it at all. It can take an hour just to get from Alverstoke to the M27 at rush hour, which is mad as it's only about 5 miles. If you live and work locally, or have a pushbike or motorbike or get to Portsmouth on the ferry then you'd be fine.
I loved living there, Stokes Bay is lovely and there's loads going on especially in the summer. It's shingle not sand but it's a beach! And it's relatively quiet without all the touristy crap you get at Southsea or the Witterings. Lots of green spaces with parks, Alver Meadow, Alver Creek. You've got Lee on the Solent, Hill Head and Warsash just up the road, the Meon Valley one side, the New Forest the other, the South Downs and the Isle of Wight.
Any questions just ask, hope you enjoy being there 