Do you drive and have a car? Don't move to Bourne if you are reliant on public transport... There is a bus service but it's very limiting.
My parents live there, I've not lived there for a - oooh - quarter of a century. So I know the basics.
There are some good food pubs in town, and some nice Indian/Chinese/Thai places. There are a few chip shops even though George's Fish Bar beats them all hands down... Loads of good food pubs in the surrounding villages, as well.
Shopping - well, there are supermarkets now (when I were a lass - it was Budgens and the market 
). Including an M&S Food... good health food shop in town, but if you are into your cooking you'll want a delivery because exotics things like fresh herbs are in short supply in town. (Waitrose at Peterborough, smaller one in Stamford where there's also a massive Morrisons). Market is a lot smaller now than it used to be but does have a good fishmonger & veg stalls (on a Thursday anyway). Good butchers in surrounding villages, a couple in town but worth a drive to Manthorpe.
Other shopping - erm... go to Peterborough or Stamford.
Doctors are pretty good I would say. There's the Wellhead park, and a couple of recs (notably, the rec).
How quiet do you want? Witham on the Hill & Edenham are villages between Stamford & Bourne.
I'd move to Stamford personally - a bit more going on and altogether nicer to look at than Bourne. (Reflected in the prices of course...)