The soap you see Barbers lather and leave on the face soaks into the hair and softens it, makes it easier to cut while also helpin the blade drag vet your skin without issue. All razors are mechanically identical (blades at an angle) and usually the same blade as well. The differences are in blade count, pivot strength if there is one, and whether they have a moisturizing strip.
At the very least, buy a dollar bar of dove soap, and slick up your legs. As for how to tend it now, well... Go easy on the skin, drink plenty of fluids and be sure to shave again soon, so the skin doesn't get the same level of stress.