This is a very open-ended question. I regret that -- I would like to be able to make it more precise, but I don't know how. I would appreciate comments on how to improve this question.
I had my first course in topology a couple of years ago and did later very little about the subject. I received a good grade then and thought I wouldn't have much trouble during the Topology II course. Unfortunately, this is very far from the truth. I'm struggling to understand almost every sentence that is said there. And one of the things that cause me the most trouble is quotient spaces.
I know the definition of course and I know that taking quotients means "gluing equivalent points together". The problem is that it doesn't give me much insight into how a given quotient space behaves. I find myself staring with my jaw dropped at people juggling with different interpretations of the same space as quotients of different spaces. When the real projective plane was discussed, I asked how I can visualize it. I was told to "simply take a sphere and glue antipodal points together." Well, fine, I understand that this is a well-defined operation on topological spaces, but the problem is that I can't visualize such a thing at all. It's impossible to do this to a sphere in the world I live in.
I've studied mathematics long enough to understand that I won't be able to visualize everything. But in this case, other people seem to have no problems with this. I would like that too. Could you please help me with it? I would appreciate answers explaining particular examples of quotient spaces that cannot be embedded in $\mathbb R^3,$ but also any other kind of answer that you think might help me overcome my problem.