I will offer a few of my favourites, but clearly these are based on my own interests:
IWM Duxford, one of the finest aviation museums in the world, if you like aircraft (and also combat vehicles) you could spend many days here and not get bored. Just outside Cambridge it is pretty easy to reach using public transport.
RAF Museum, Hendon and Cosford sites. The Hendon site is in North London, Cosford is on the site of RAF Cosford just outside Telford (near Birmingham). Both of the sites are superb, I prefer Cosford now as they swapped some of the exhibits and Cosford got a few things that used to be in Hendon. Free entry, both are easy to reach using public transport.
The three museums in London Kensington, the Victoria and Albert, Science Museum and Natural History Museum. All three are pretty much next door to each other, and free entry. The one I'd recommend you check out and which many ignore is the V&A. The V&A is a museum of fashion, jewellery, ceramics, glass etc, it's a museum I thought would be boring but it is truly fascinating and I could spend many days in there, the collection is remarkable.
Also in London, the British Museum and the national Gallery are within easy walking distance of each other and free entry, the National Gallery is in Trafalgar Square which is an iconic location and feeds into both Whitehall and the Mall, one if an interesting walk up to Parliament and Big Ben and Westminster Abbey, the other a lovely walk up to Buckingham Palace.
Portsmouth Historic Dockyard, this is part of Portsmouth naval base and home to HMS Victory and HMS Warrior amongst other things (the Victory is still in commission and still has a commanding officer).
Blenheim Palace, near Oxford, if you like stately homes and palaces then this is the grandest of them and well worth a visit.
If you want to see lovely country then I'd recommend Snowdonia, the Cotswolds, Lake District and Yorkshire Dales national parks. Snowdonia and North Wales is a wonderful place to visit with loads of things to see and do, and not that far from Coventry. You can do these with public transport but it is much more convenient if you are happy to hire a car.
You will be close to Stratford on Avon, bit touristy and a bit of a Shakespeare olde world theme park but nevertheless worth a visit. Oxford is a nice city (well the centre is). There is actually a very good motor museum in the centre of Coventry, and it is free to enter. The national museum of motoring is also close to Coventry.
For music, Coventry is close to Birmingham for the CBSO, while there is more good live music in London (of all genres) than you can shake a stick at.
For a short day trip from Coventry, take the train to Bletchley and visit Bletchley Park (it is just over the road from the railway station, frequent trains) where Alan Turing worked and the Bombes broke codes.
I could go on, but these are just a few things I'd recommend.
