Talk to your swim instructor about the trouble you are having with breathing. I'm sure there are things he/she can do to help you. I used to teach swim lessons and I would have my students practice breathing with their feet on the bottom of the pool while holding onto the side of the pool. It's a good way to get a feel for the motion without the added stress of trying to float at the same time. Once they got that down, I would have them add floating and kicking to the mix (with breathing, still holding on to the side of the pool). Then we would move on to actual swimming. Of course, this was 20 years ago, so your instructor probably has some new tricks. It can take awhile to get the motion down and to feel comfortable with it.
Swimming is a great cardio exercise because you have to hold your breath while you do it. But having to hold your breath can also make it tricky to learn; it can take awhile to get comfortable with it. Often times people will try to push themselves too far out of the water when taking a breath or turn their head too far, especially if they are afraid of swallowing water or not getting enough air. It's impossible for me to know without seeing your stroke, but this might be why you are feeling a strain in your shoulders and neck.
Swimming also works muscles that we don't normally use (another reason why it is such a great workout), so you might just be sore because your muscles aren't used to the workout yet.
You also might want to try a different stroke. You might be more comfortable with breast stroke (where almost your whole head comes out of the water), side stroke, or back stroke (no trouble breathing there). Eventually you'll want to master the crawl if you want to swim for cardio exercise, but you might be better off starting with a different stroke until you are comfortable in the water.
Everyone learns to swim at their own pace, the biggest factor being how comfortable you are in the water. Adults were at the same time my easiest students (because they could understand what I was telling them to do and easily apply it--5 year olds just cannot do this) and my hardest (because they had a lifetime of fear of the water ingrained in them). If you are comfortable in the water, you'll probably learn very quickly. If you are working through some ingrained fears, it can take a little longer.
Stick with it and I'm sure you'll like it. I am a complete klutz and miserable at most sports. Swimming is the only sport where I have ever felt confident, graceful, and in control of my body. Unfortunately it ruins my hair, so I don't do it much these days.
- Barbara