One of my friend's is in potty training H-E-L-L right now. After giving her a pep talk because her little girl had 7 accidents on day one, I remembered that I carried a Potty Kit everywhere we went while my son was newly sans diapers. I took it to Target, the art museum and on the airplane when we flew to California for a wedding. It helped tremendously - and what I mean by that is it helped my son go the bathroom in public, and it helped me be at ease during this stressful period!
Washable, water-resistant, lightweight bag. Our Envirosax bag worked perfectly. Its nice to keep all of the below items in a bag while traveling. Keep it separate from your diaper bag and especially when at an airport. I would put it in my carry-on, and then grab it, take my son to the bathroom and leave the carry-on with my husband.
plastic grocery store sack (for a dirty diaper, or wet wipes, etc.)
Change of clothes and undies
Diaper (just in case!!) and wipes
Diaper cloth or wash cloth (my son peed all over a sculpture garden outside.)
Mini toilet ring stored in a Ziploc bag (it can get gross if you are using it all day on a trip or in an airport, so keep it in a gallon-sized ziploc.)
This is the one we had, it folds into a 1/4th of the size shown –