How to clean up after kneading dough?

Alternatively (with respect to @GdD's answer), let it dry out completely. Totally dry dough doesn't stick all that well to many surfaces (glass, plastic, non-stick). It then chips/scrapes off quite easily. If I get it on my oak worktops and don't notice immediately, that's what I do, scraping with a plastic scraper or a butter knife, even a fingernail on the last bits; the same approach works on my stand mixer bowl and dough hook. With some containers you may need to soak a few stubborn bits especially if they're caught in corners, but oiled wood doesn't like that.

If you are clearing up wet, very wet is the way to go, and a brush clogs less than a sponge.

I hand-work my dough either on a pastry mat or if that's dirty non-stick sheet (which isn't non-stick against dough unless well-floured, but cleans easily). These are both fairly easy to wash up.

Kneading in a bowl is time-consuming and doesn't give as good a result as kneading on a flat surface, however I'll concentrate on cleaning.

First, don't let things dry out, it's much easier to clean when things are moist, if you do let it dry out moisten it and let it soften before you try and clean it. Use cold water as hot water makes starches and proteins stick a lot more. Next, invest in a curved dough scraper, and use it to scrape out the dough scraps from the bowl into the garbage before you clean it. A curved plastic one works best as it has a bit of flex, although in a pinch you can use a big metal spoon. I scrape by bowl pretty much clean before I put it into the sink. Use the flat of the scraper to clear most of the dough and flour from your countertop, then spray down with water, let it soak for a few minutes, then scrape again. Once you have it scraped well sponge it down.

Amazed nobody else has given the easiest answer, which is to soak it in warm soapy water for a while (at least a few minutes, up to an hour) before scrubbing it out. It makes everything soft and partially dissolved and it just wipes away.

For handwashing: Use a dishwand instead of a sponge (the dough won't stick to the plastic bristles). Use warm rather than hot water, to avoid 'cooking' the starch onto things - but no need to use cold water.

Alternatively: Scrape as much of the dough as you can into the bin with a plastic spatula. Then put it the bowl in the dishwasher.

and any sponge I use for it gets ruined afterwards

I had the same problem until I started to use a brush like this one

_{(courtesy of Ikea https://www.ikea.com/fr/fr/p/rinnig-brosse-a-vaisselle-vert-90407811/)}

It really makes a difference: cleaning the brush is much easier, including greasy/oily substances (in addition to sticky substances like dough in your case)