I'm glad you didn't wash dishes in the stream or lake as some beginners do. Contrary to popular belief, fish don't eat the food particles; the food just rots and degrades water quality. Not something you'd want to leave for the next hiker's drinking water.

The old-fashioned camp dishwashing method that I learned at my mother's knee required a washbasin. We horse-packed, so weight wasn't so critical. A kettle could substitute. She washed the dishes in a small amount of soap (not detergent) in really hot water, and I dried each item as soon as she washed it, without rinsing. The idea was to wipe thoroughly to remove any soap. I don't remember anything ever tasting soapy. The soapy water was, of course, dumped a long way from the campsite and from any water source. It is recommended that you filter the dishwater through a piece of paper towel or a coffee filter to remove the food particles, which you pack out.

Your wet-wipes sound like a great idea for a short trip, and I plan to try it! The weight of the wipes would be an issue for long trips, though. You could let the used wipes sit out for a few minutes to dry afterwards to reduce your garbage weight. I hope you're doing the final rinse with filtered or purified water. This just shows that beginners can be more creative than us so-called "experts"!

I avoid washing dishes as much as possible by using the Freezer Bag Cooking method. Nothing to wash except my spoon, which I lick, then rinse and dry. If I have a hot beverage, it's tea, so I just rinse the cup. I use my leftover cooking water for this operation. When my grandkids come on a trip, I have to wash a lot of dishes. (Did I mention that I hate washing dishes?) For next year I plan to make them individual cozies so I have a freezer bag for each child. I'll still have to wash their cocoa cups, though.

I assume that when you said "thrown away on the way out of the park," you meant deposited in the park's garbage can. I prefer to take my garbage home for disposal--a reaction to seeing too many park/forest/roadside rest area garbage cans overflowing after a weekend.

Starting out backpacking by going only a mile or so in is also an excellent idea. It sounds as though you're off to a great start!
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May your trails be crooked, winding, lonesome, dangerous, leading to the most amazing view--E. Abbey