I have two ostomy kit bags that were gifts from vendors. They seem to be modeled off of men's travel toiletry kits. I like to keep them in the car as they are big enough for some extra supplies.
Over the years as an ostomy care nurse, I have seen two schools of thought on this. One is to bring only what you need to clean yourself up enough to get home and then do any major changes.
The other school of thought assumes that you are going to continue on with life and not allow this to jerk you back home every time it occurs. Those people need to bring just a little bit more.
First, of course, the means to clean yourself up. I will do a separate post on possibilities for cleanup. I do not recommend baby wipes or diaper wipes as most of them have some type of barrier product such as silicone or lanolin or other emollient that once it is on your skin it will inhibit your next appliance from sticking. Certainly a change of clothes because it isn't just the lower half that can get baptized in shit if you have a longer tunic or shirt in the interest of covering up your stoma and your appliance secondarily to that a complete change just as if you would have had to do at home. If your appliances are cut to fit, you will need some small scissors of some sort. I recommend curved; they come in those aforementioned kits from some vendors.
NOTE: Depending on the price, if your scissors are not sharp, either a) sharpen them or b) replace them.
For me, a quart-size Ziploc bag will hold the scissors and one wafer (aka base-plate) and two half-moon-shaped barrier extenders and one Convatec "ring" called stoma wrap and one pouch (aka bag). A separate bag holds my clothing and clean-up supplies.
This is everything I need. It is identical to what I would be putting on at home. However, it means I will be spending some time either in the handicap stall or even in my car.
If friends or family are waiting for me, say in a restaurant or at an event or a movie or park, it is important that they understand the approximate time frame it will take you to clean yourself up and get changed and get back to them, whether you follow plan a) run home or plan b) accept this and move on. You will still be returning to the group you left behind to change your appliance, and it is important for them to understand approximately how long you will be gone. They tend to get upset and even a little panicky when you are gone for that length of time, particularly in a bathroom, as if they thought it was the rudest place on the planet.