In my experience the different branches use relatively the same type of equipment, but different branches like to specialize. There are exceptions but they tend to be mission specific. The AF flies cargo planes and bombers, the Army has their helicopters, but most branches dabble in fighter jets (with the exception of maybe the Army, I’m not sure about them.)
All the large “people movers,” were usually commercially chartered (not as cracked up to be as it sounds, I’d rather be crammed in with cargo than crammed in with people any day.) until they arrived in theater and then they were often military aircraft. Basically, they would use commercial charter to shuffle us back and forth over the ocean but use Military aircraft once we got in the general area where we were going. I was a dog handler so I often found myself in with the gear on planes like the C141 (yes, now retired), 130, 5, and 17, all of them AF planes. If you want to get up close an personal with one as a civilian you could try visiting your closest air base, most have air museums (of the actual planes) that you can visit.