Asics running shoes are fine shoes, so long as they were made for running (vs. basketball), and do be sure they aren't older than a couple years, they will degrade over time. I get new shoes about every 2 years and I get about 500 miles on each pair.

Like a couple others have stated, barefooting is fine if you have the gait and body type for it, most of us don't. Best I can suggest if you want to try it is find a grass park or beach to experiment on, and work your way into asphalt. If you can comfortably do a mile or so then look into those faddish Vibram Five Fingers.

To directly answer your original question - some of your discomfort is due to lack of use and will get better over time as long as you don't push too hard. Skin is not the only body part that develops callouses - your bones, muscles and even tendons need to adjust to the new routines you are pacing them through.