So the Vatnajökull ‘tour’ you mention is cross-country skiing across the ice cap, a several day expedition. Not really the correct terrain if you’re looking for fun touring up mountains etc. There is touring in the southern and eastern regions, plenty of it, but December is difficult because of light (few hours per day) and possible lack of good snow (maybe better in later winter). That said, it’s not impossible and some good days can still be had.
http://www.wikiloc.com has plenty of info on backcountry routes people have done and is a good starting place for figuring stuff out.
There is also plenty of good day/few hour long tours close to Reykjavík if the snow allows.
If you want to rent gear the only places are in Reykjavík really, and it’s not cheap.