+1 for Gosforth. If you ever fancy a bit of raod riding you're straight across the A1 and into some nice countryside that they use for the Northern Rock Cyclone, or my little ride past the Cheese Factory Farm and around the fancy houses in Tranwell Woods (and no, that's not a wood to ride in, its a place where footballers buy their houses if they don't fancy Darras Hall).
Be careful though. What people call 'Gosforth' exetnds waaaay past the boundaries of what Gossie really is. See many a Kenton place described as in the Gosforth area.
If you don't mind being close to Tescos, Kingston Park isn't too bad.
And for a 'secret' location that's a little cheaper – try Cochrane Park, just off the Coast Road, just up from High Heaton. Decent buses, Sainsburys etc. though not many rentals there I don't think.