I think you are on to the right mindset for a beginner. Don't worry about distance so much a time in the saddle.

I don't know what your neighborhood is like for riding around in but just riding around exploring, taking it easy and enjoying the breeze in your face is how you'll start building a base.

Some lightly padded gloves may help with the tingling but a really proper fit is the best way to combat hand pain. When your weight is correctly distributed you will put less on the bars.

Here's a link to Sheldon Brown's Beginner page. Good Stuff! http://sheldonbrown.com/beginners/index.html