In reply to mostly harmless: I spent last week in the Jura, mostly day walks although some sightseeing as my partners not able to walk much at the moment. The most alpine part is the Crete du Neige which I walked up from Lelex & took a detour to the slightly lower summit of the Reculet. I'd recommend this as a good day with (if you are lucky) views over Lake Geneva to Mt Blanc. The cable car from Lelex would make this much shorter but I think it only operates at weekends in the summer. There are also some obvious walks from the nearby Col du Faucille which you can drive to.
On a more sightseeing theme, the saltworks at Salin are worth a visit
The guide book I've used for visits is Michel Blant's "Le Jura - Les paysages, la vie sauvage, les terroirs" but its a guide to the natural history rather than a walking guide.