I recently learned to do the front crawl. I am following the advice I've read online about exhaling the entire time my face is under water, and relaxing. Regardless, pool water is choking me, coming up through my nose, and running down my throat. Also, though I am trying to relax and not rush things, once in awhile my heart is beating very fast and I am very out of breath after swimming just a small distance. What might I be doing wrong? For perspective, I am a long distance runner and I don't think I have any fitness issues that would cause me to be so tired after such a small swim. I mention both the being out of breath, and the water in my nose and throat, because I think they may be related.
It is difficult to say without seeing your swim stroke, but I suspect that when you are breathing, you are over rotating, with your head pulled back rather than being in a neutral position. That would allow water to enter in your nose.
When swimming freestyle, you don't really move your head much. The top of the head should be in the water, which will create a little bit of a "bow wave" effect, and you breathe in the trough that is created. There is almost always a little bit of water that can enter into the mouth, but you generally exhale it out with the breath.
And while you may be in good running shape, swimming involves many more muscles (especially upper body) that aren't used to the effort. Also, it's very technique driven, so if your technique is poor, you will fatigue faster as you "fight" the water. I would recommend an adult beginner swim program, it will help immensely. If that is not feasible, look at programs online such as swimsmooth and Total Immersion.
You should try a technique involving moving your soft palate into the right place by saying the letter 'T'. Try mouthing the letter and you will see that you cannot breath through your nose, so water will not get in to choke you. Here is a more detailed explanation: http://blog.swimator.com/2011/04/getting-water-up-my-nose-while-swimming.html
Exhale through your nose, too, in addition to the other advice. Even if you're swimming fast (or, rather, particularly then) you can exhale both through the mouth and nose. Benefits are at least twofold: you get the air out quicker, getting ready to inhale properly, and the air going out of the nose keeps you nose nice and comfortably not waterlogged.