I was just asked a similar question not too long ago via a private email. This was my response.
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Recognizing musical counts is a very involved topic, and not one that I can answer easily in an email. There is no "magic" answer that will solve your problems... like learning to dance, it's a skill that takes time to develop, and the development is usually a slow and gradual process. Some people have more of a natural affinity for it than others, but the good news is: Like dancing, anyone, even those with no natural ability, can learn with enough time and effort.
Dance teachers are not always the best suited to really teach music. Don't get me wrong... they do have a good deal of knowledge about music. But to really teach someone a concept as rudimentary as feeling a particular count (1 vs 3, strong measure vs weak measure, etc), it takes a true specialist. If you're really dedicated to learning such concepts, I'd suggest taking some lessons from a music teacher.
One of my proudest accomplishments in my dance career was teaching a particular lady who had so little concept of rhythm that she couldn't clap along with a metronome click. This was just about the most un-musical person I had ever met, and of all things she wanted to learn Salsa. But after 6 months with her I had her clapping along to actual Salsa music and finding the 1 beat. Now I do have the advantage that I also happen to be a music teacher, but more to the point, I would say that more than 75% of the first 10 to 20 lessons was spent standing in place and clapping or marching along with either a metronome track or some simple piece of music. In other words, I literally gave her dozens of hours of basic music instruction, not dance instruction. That's the kind of dedication it took, and if it weren't for her persistence, she would never have learned. Most people lack that kind of patience and persistence (most would have said "forget it, let's dance!" after the first 10 minutes), but like I said, if you really want it, it's not beyond your reach. However it may take some serious focus on music, without the distraction of dance concepts.
Now again, learning how to find the basic pulse is a much easier task than learning how to find 1, or understand the difference between 1 and 3, strong and weak measures, etc. But that, too can be learned just the same.
[Snipped out a few paragraphs about finding beat 2 in Mambo]
Finally, keep in mind the cardinal rule of music instruction: There is no better teacher than your own two ears. So start listening... a LOT. If you start with stuff you recognize, you will have a seed which can slowly grow roots and eventually blossom. For example, "Happy Birthday" is a song we all know. If you know in advance that the word "birth" is the 1 count, then you'll always know where 1 is in that song. Without lengthy explanations about phrasing and harmonic transitions, you have now begun to develop an intuitive sense of what makes a one a one! Now take 25 more songs you know well (and I mean the kind you know so well that you sing along with 'em end to end) and do the same thing. Can't find 1? Ask someone. Make a note of what lyrics fall on the 1. From now on, you'll not only know where the 1 is in those songs, but in any music that is similar. The more songs you know, the better you get, and the better you get, the less similar one song needs to be to the next for you to be able to do it on your own. And in the end, there is very little music that is truly original... most everything is like something else. So you may reach that goal sooner than you think...
