How do you swim and breathe at the same time?

I am now swimming twice a week on a fairly regular basis and I’m feeling fitter for it. I’d like to go three times a week but there just aren’t enough hours in a seven day chunk, so twice a week will have to do for now. All I have to sort out is my breathing.

I’m one of those swimmers who do breaststroke with my head up, like a swan, only nowhere near as elegant. I watched how the other swimmers were doing it. (You have to be careful staring at other people in a swimming pool. You run the risk of misunderstandings.) What I needed, I decided, was to buy some goggles. This I did and on my next visit I enthusiastically pulled them on and launched myself into the water but that was when I realized that I didn’t know how to breathe. I tried not to attract attention as I spluttered and gasped and I resolved to find myself a swimming tutor.

This is turning out to be harder than expected. There are adult beginners lessons but I’m not really a beginner. Although I can only manage one length of front crawl (probably because I don’t breathe properly!), I can swim breaststroke for many lengths, so I’m now trying to get some one-to-one lessons. To date I’ve had no luck.

There must be someone out there who can teach me how to swim and breathe at the same time!

p.s. In response to Jo's comment below, I realise that I forgot to mention the neck ache. That's my reason for wanting to learn how to swim properly.