How long can I keep mushrooms in the fridge

Mushrooms are mostly water, so you need to keep them from getting any wetter. If you buy them in those strange material boxes wrapped with cling film, or in a plastic bag, remove them from their container when you get home and put them into a brown paper bag. I put the date on with a Sharpie.Don't put them in the crisper, as that is generally more humid than the rest of your fridge. I find they usually keep about a week stored that way. I rarely if ever buy sliced mushrooms as I find they spoil more rapidly than whole ones.
I toss them without delay if they become slimy. If they have withered and dried, they can be used as dried mushrooms, or reconstituted for immediate use with water.

If I need to prolong the life of mushrooms (5 or so days in the fridge seems max for me) I will sauté them and that will give them another several days of life. Great to have on hand to throw in an omelet or on a burger.

If you keep them in a (preferably) cloth or paper bag they will slowly dry out, but still be quite usable. In plastic, they are likely to turn to soup. Generally, they are a little past ideal when the caps start to open (at least the common supermarket types of mushroom)- this usually only takes 3-5 days or so, but they're still good to use.

i buy mine in a large plastic container wrapped in plastic as costco. i remove plastic wrapping, rotate bottom mushrooms or any that have moisture on them to the top, and cover the box with two wetted and squeezed out paper towels. they tend to keep for two weeks or so if you keep the paper towels moist and rotate any moist ones out and use them.