What benefit do you think new rims will provide?
Fast and puncture resistant tend to at odds with each other, best have a look over sheldon brown's website for general information. Your requirements are so broad it there's no clear recommendation to be made other than some variety of Marathon tyre, i.e. a general purpose touring tyre with decent puncture protection.
But really you want to start with your bike as it currently is: how do you find the ride in terms of comfort? What psi do you run your current tyres at?