I have an issue with breathing while strength training. Whether it’s pushups, sit-ups, crunches, or free weights I have trouble finding the correct breathing rhythm and wear out quickly. I would appreciate any advice.

A couple of questions --
-Do you try to breathe out on exertion, and if so, what happens? -Are you going too fast to get into a breathing rhythm?

There are lots of articles on breathing during exercise. Here's a short one.
http://www.military.com/military-fitness/workouts/breathing-during-exercise

I try to inhale at the point of maximum exertion, but some times I catch myself holding my breath.

And for the sake of clarity, this is only a problem when strength training. I have no problem breathing when doing cardio or fighting.

My first reaction to this is to suggest that for a while, instead of doing strength training and worrying about breathing, reverse it -- concentrate on BREATHING while doing activities like strength training (in other words, make the breathing the focus, and don't concentrate on improving the strength training until you're satisfied that the breathing is supporting your rhythm properly).
If you go back to old habits, focus on the breathing again INSTEAD of the strength.

What do you think?

Check out the book "Let Every Breath.." by Vladimir Vasiliev. You will find numerous breathing-body coordination drills to help train your body.

Take care,

Mark J.