I like to be leisurely when eating apples. I usually use my pocket knife and just pare off slices from the whole apple, eating them one at a time, sometimes with a piece of cheddar. Once I’ve gone around the apple a couple of times in this manner, I just gnaw off the remaining flesh and discard the core.
Another vote for the square thing. It’s definitely the easiest way.
That’s how I slice apples for cooking, too. I stand it up, cut off one nice round slice, then work around in a circle, angling the knife to get more-or-less even pieces. Then I trim all I can from the core. I never ever cut apples into thin pretty slices - I like my apple desserts chunky.
I’m probably going to lose a finger doing it this way, but it’s how I’ve always done it.
I cut the apple in half from the top. Then I gouge out the tough stuff and seeds. and make a couple of more slices to get out the rest of the core. Goes pretty quick with a paring knife. If I need to peel it I’ll shave it with the knive with strokes from top to bottom.
Some more details and an update.
I wanted very thin slices to go with some cheese and crackers. I have one of those things you press down on the top and core and slice it into eighths or whatever, but it is just something else to wash when I usually already am using a knife. Plus, since I want the thin slices I’d have to use both that thing and the knife.
Tonight I decided to try the square core technique since it was mentioned so often. Meh… the slices just didn’t look right for display. Plus it looked like I was wasting so much of the edible parts around the core.
Then I remembered this…
This my friends is true genius. Cut it in half, scoop out the center and perfect. Sure, I just contradicted myself on one more thing to wash, but the melon baller is easy and quick, and much easier to clean.
I know, right? And, yes, if you’re handy with a knife it’s entirely possible to do the same scooping out of the core with a knife. My knife skills just aren’t quite up to the specs my OCD requires, and the melon baller makes such a beautiful, clean, symmetrical cut. 
I do use the apple corer slicer sometimes, mostly at my daughter’s request 'cause she thinks it’s neato. Like you, I hate washing it. I also hate the mush it makes of the cut edges of even firm crisp apples, the increased rate of browning as a result, and I hate lining up the core just so, and I hate hate hate having to finish each cut by poking the pieces through with my finger 'cause the skin on the bottom of the apple won’t be cut by the slicer.
Don’t use a melon baller use the same paring knife that you used to cut the apple in half. Stick the knife in to the apple so the tip goes underneath the core and the knife exits at the side of the core. Then make a circle with the knife, basically cutting a cone shape out of the apple which contains the half core.
Doesn’t use any extra tools. Try it!
Huh. My apple slicing technique is totally different, it seems like:
- Slice the apple into vertical quarters, going straight through the core.
- Pare off the core from the sliced apple
- Cut each quarter again lengthwise.
- Eat.
If I’m processing an apple for cooking, I almost always use a mechanical spiral peeler/corer/slicer.