The lanes around Balsall Common are generally quiet if you don't go too close to Coventry. Go north,as Fungus suggests, and there are hills with views around Maxstoke, though no challenging climbs.
Go south and there are quiet lanes through gently undulating farmland with hedges, many oak trees and more hamlets than villages. A circular ride would involve a short section of either A4177 or A4141 - not ideal, but they're roads I ride on regularly. Suggest Honiley, Beausale, Haseley, Baddesley Clinton as a route than can easily be shortened or lengthened according to taste & weather.
As to pubs...
At the further end of a loop, the Griffin in Church End (Shustoke) is a truly excellent watering hole to the north, while The Case is Altered in Mouseley End is equally good to the south (warning, there's a £1 fine for anyone whose mobile phone rings inside this pub).
Closer to Balsall Common, the Bull's Head in Barston is a personal favourite, closely followed by the White Swan in Hampton-in-Arden & the Tipperary Inn at Meer End. The Railway Inn in Balsall Common, by Berkswell station is also good, & might prove more convenient.
I'd like to join in an evening pootle. Balsall Common is only about half an hours' ride from home.