I have a couple stainless steel skillets, one of which I use several times a week for eggs. I wash them by hand with soap and water after each use, which is not something I would do with cast iron. If I cook something that gets the skillet really nasty, I simmer some water and baking soda for a while, and then while it's still on the stove I use a silicone spatula to scrape off all the stuff that's stuck to it. This has worked well enough for me for as long as I've done it.
I know some people do season stainless steel, but I've never tried it myself. I do cook with coconut oil, though, so maybe they've kind of self-seasoned over the years.