You could spend a week in this area and not run out of things to do!
Newcastle is quite hilly, so stay around the rail station if this is an issue. It's always useful to think in terms of local currency when querying accommodation prices. You are currently getting lots more pounds for your dollars.
You don't say when you are visiting - some things are closed late autumn/winter.
The train to Durham is quick and easy. Durham too is quite hilly, with steep cobbled streets. The cathedral is magnificent, but I have always found the rest of the city to be a little disappointing. It's a half day trip from Newcastle. Have you considered visiting Newcastle Cathedral instead? It maybe better use of your limited time, with so much to see in the area.
There are several National Trust places in the area that are very interesting. You can get short-term overseas membership. Hadrian's Wall and Housesteads Fort that they own is worth a visit. Their website gives directions as to how to get there - sorry, my mouse won't allow me to post the actual link, so try National trust.org.uk. Their Washington Old Hall nearby has connections to George Washington that you may find interesting. One of their most interesting properties in the country IMO is Cragside in Morpeth, the birthplace of hydroelectricity in Victorian times. You probably don't have time for the brilliant Beamish Museum, as this is at least a full day trip.
There are walking tours in Newcastle - a good way to see the city. The people there are the friendliest in the UK, but the accent can be difficult to understand! "Whey aye, pet, she's bonny" (pronounced why eye) means "yes dear, she's pretty". Don't let this put you off! You said you wanted English Life....