In reply to munri:
Hi
I don't know too much about campsites, nor the running or biking, but I would say that for climbing in May, I would base yourselves near one of two epicentres of gritstone climbing: either Stanage or The Roaches. These are far and away the biggest and best crags, and if you're camping right next to either, you're also in walking distance of lots of other classic crags.
I would say that the limestone in the Peak isn't really worth travelling to specifically (although I expect someone will recommend you go to Stoney - don't!), but if it's really hot you might find that everything on grit feels impossible (the friction-dependent nature of the climbing means that in hot humid conditions the climbing is really difficult and not very fun). In which case, if it would be worth checking out the limestone, bearing in mind that at HVS and below you'll encounter a lot of polished horrors.
As for pubs, if you're near Stanage, then Hathersage has some OK pubs, but I particularly like The Grouse at Froggatt edge, which does a great steak pie - the best in the Peak by a country mile. Avoid the Fox House near Burbage, it's dreadful. If you are out on the limestone, then the Red Lion in Litton is my favourite - in fact, I'd go there for a meal and a few beers anyway, it's great. There are also quite a few posh gastro-pubs if that's your thing, but I tend to avoid them (although I have had some excellent food at The Plough in Hathersage).
Others will know the pubs around the Roaches area better.
Hope that's helpful.