Earlier quoted context omitted.
I recently switched from teflon to stainless steel, and after a few false starts figuring it out I like it more now. Also for something that leaves a lot of grease like sausage and bacon, the stainless steel surprised me by being easier to clean.
What did the false starts teach you? I have some stainless steel pans and sometimes they work great and sometimes they're just mysteriously sticky, with whatever I'm cooking bonding to the pan.
* You should preheat until you get a leidenfrost effect - get some water on your fingers and flick it onto the skillet, it should form droplets that bounce around without boiling off.
* Once this happens, immediately turn the heat to the lowest setting so the skillet doesn't get too hot (most instructions are missing this step and I didn't really expect the skillet would get too hot - stainless steel just keeps getting hotter at a temperature the teflon nonstick would maintain heat). Depending on your stove, the lowest might be a little too low and heat is slowly lost, this is something you'll eventually get a feel for.
* After getting the leidenfrost effect (maybe? not sure how important it is to wait until after), you have to add something (butter or vegetable oil, for example) to coat the surface and not only get a nonstick effect but also sort of buffer the heat. It's kind of like cast-iron seasoning, but extremely low-effort and you do it / clean it off every usage. Some of the things I'd read before buying said the leidenfrost effect was the important part for getting a nonstick surface without mentioning this stage, which led to a pancake that was black and stuck to the skillet on one side and still liquid batter on the other.
And when cleaning:
* I'd mentioned sausage and bacon above, these leave gunk that sounds kind of like what you described except a lot more of it. For some reason getting a wet paper towel and rubbing down the skillet (instead of putting water on the skillet directly) works really well getting almost all of it off, though it will take several of them.
* If there's still residue not coming off, something I got from reddit worked even where grease-removing dish soap didn't: Lightly boil baking soda in water in the skillet for about 20 minutes. Don't let the water boil off or you'll be left with baking soda gunk stuck to the skillet, you want it to dissolve and soak in the hot water for a while before emptying it and wiping it down.
* And lastly one of the side reasons I like it over nonstick while cleaning: The surface is actually smooth. When using a scrunge to wipe it down you can feel where there's still something stuck to the surface, while nonstick is rough even when clean.