I am learning to swim, but my biggest problem is breathing; not only does my heart rate rise rapidly, but I get out of breath quickly. I have read that proper exhaling plays an important role, but how do I know if I have exhaled completely? I workout 5 times a week, and I am indeed fit, but in the pool it seems like it's an entirely different challenge. Also when I turn my face to breathe, I start breathing from my nose. How do I develop the habit of breathing from my mouth? How different is it to breathe on land vs. water? How long should I keep my face turned to breathe? I can barely turn my face for a second, I guess, so how can I breathe in such a short span? Even if I breathe, it's insufficient. Is there anyone who suffered with the same problem while learning to swim? If yes, then how long did it take to overcome that issue, and how?
I understand your conundrum. In a way, being fit and having good cardiovascular capacity makes finding your pace a little harder.
Exhaling. Begin exhaling while your face is still in the water and blow out as much air as possible. Your goal is to have your lungs empty so that when your mouth and nose break the surface your body is really wanting to breathe in.
Your pace dictates how long your face is out of the water. Your body is rotating on the long axis and your face breaks the surface as a result of that rotation. If you find that period of time is too quick to get in a decent breath, then slow your arms down and exaggerate the rotation.
Breathe in as is most natural for you: nose only or nose and mouth. If you want to force using your mouth, there are nose clips available.
You may (probably will) find that your body is up to a faster pace than your breathing allows at first. If you focus on form, particularly rotation, rather than speed the breathing becomes more natural.
I found that a kick buoy helps so that I can focus on the upper body. Rotating in one direction only for several laps helps too, that means turning your face to the same side to breathe in. After a few laps always turning in one direction, switch and rotate to the other. Rotating in one direction will feel natural and the other will feel weird, but that makes the rotation even so work on both.
I spent perhaps more time than I should have this morning during my swim thinking about your question. I think that there may be a number of different things to think about here.
On getting out of breath quickly: (1) you are learning to swim, so it is all new, particularly having your face in the water. A little adrenaline and fear goes a long way towards getting out of breath quickly. Novice SCUBA divers are well known for rapidly emptying their tanks since they are not feeling calm, cool, and collected underwater. (2) Swimming uses all of you large body muscles - arms, legs, and core. You need lots of oxygen to keep all those muscles going, much more than running (in my experience). And, unlike running, you can only breath at certain times. As you get better at swimming, and more comfortable, and in better shape, your breathing will get better. Swimming well is doing lots of little things properly, and you have to intentionally focus on each little thing, make it better (and automatic), and move on to the next, eventually coming back to the first to make it even better. Breathing is, of course, key to being able to swim, so it is something that you will have to focus on doing properly.
On breathing through mouth or nose: Every competitive swimmer that I have ever met breathes in through their mouth, and out through the nose or nose+mouth (when you really need to get air out fast). You just don't have time for leisurely breaths. As an aside, you will take in water from time to time - learn to deal with it. And, it is much easier to deal with water in the mouth than going up your nose.
Finally, I will disagree with @Val above on one key point. As a long time swimmer and swimming official, the biggest (freestyle) stroke mistake I see in newer swimmers is over rotation of the upper body during the breath. This over rotation is probably caused by the swimmer trying to get the mouth and nose well clear of the water before breathing. This really messes up the shoulder position, and will drag the legs around as well, which then destroys any chance of a proper kicking action. It is a stroke efficiency and effectiveness killer. Don't go there.
If you watch experienced swimmers, you will see that they breath while the head is looking almost parallel to the water surface, with at least part of the mouth seemingly 'under water' in the wake they are making. Yes, every once in a while you will get some water from a wave - stay calm and swim on! Further, while the idea of just letting your head rotate with your shoulders is not bad (if the shoulders aren't over rotating - don't breath facing the ceiling!), the theory over the past decade or so in coaching is to turn the head to breath, then return to the neutral position, faster than the arm rotation. Keeping the head in the neutral position helps keep the body aligned better, keeping the balance of the stroke.
Your shoulders rotate to increase your arm reach forward. Your hips will rotate a little as well, but they should not rotate anywhere near as much as the shoulders (this is where your core gets a good workout!). You legs should not rotate much if any at all. An experiment to see how much your shoulders should rotate is as follows: Stand with feet flat on the ground. Keeping your shoulders level, reach upwards with one arm. Now, pretend you have to reach just a bit more to touch something above you, but don't lift a foot off the ground (you can let the heel come off the floor, but keep the toes touching). You will find that you rotate your shoulders, and the hand goes higher. This is the rotation you are looking for. Anything more is too much.
I fully agree with @Val that the ability to breathe to either side is a real asset. I started early in my career since I found I was getting side cramps on the one side I was breathing to. Always breathing facing one side of the pool cured it, and made me a better competitive swimmer since I didn't care which way I breathed - I could keep watch on a competitor, or just breathe when I needed to, without affecting my stroke. I still do it, many decades later.