I think many of us could go on for days about this. I think it's something that you have to gradually open your eyes to. I had a "cringe moment" in college where I asked a black friend if he had experienced racism. He looked at me incredulously (as he should have) and asked if I had experienced sexism. I said not really, and he just shook his head. Now looking back on that (6 years ago) I can't believe my ignorance. I've read a lot more and thought a lot more about it, and I've come to see how things like jokes and pop culture and even the science of gender reinforce day to day sexism. And I see the harm in all the simple things I used to accept or laugh off. It's so accepted in our culture that most people don't even see it, but once you do it becomes mind-blowingly obvious.

And the thing about stereotypes is that they're often contradictory. One of my professors once did an exercise where she wrote pairings of words on the board and you had to assign a gender to them. One of them was "dog/cat". We all said male/female. Then she wrote "poodle/lion" and flipped it. It always depends on context.

As another example, you mention the stereotype of women being illogical, but how many shows and movies do you watch that feature a bumbling hero who's kind of a moron but knows how to relax and have a good time, whereas his wife or girlfriend is more rational, but a little up-tight and often controlling? It's the Homer/Marge Simpson dichotomy. Or Chandler/Monica. The list can go on.

This flexibility is part of what gives gender stereotypes their staying power. They can be adapted to fit any situation. It's also part of what makes feminists seem crazy to people who haven't yet opened their eyes to these issues. Debates like the one you're talking about are so great, because they make you formulate an argument and really think all the way through these issues. Reading a lot is the other thing that really helps.