Breathe
however you need to in order to get oxygen in.
This goes for your mouth, nose or a combination of both.
Riders warming up on their trainers have cotton in their noses that contains something like vapor rub that helps open up the nasal passages.
They don't have it in their noses during the race itself.
bicycles.stackexchange.com/users/4481/JohnP
For some people, weather conditions (specifically: temperature and humidity) make a big difference.
Not so much for direct intake of oxygen, but because repeated inhalation of cold, dry air directly through the mouth can cause significantly more irritation of the airways and lungs , triggering asthma symptoms.
Breathing through the nose warms and slightly moistens the air before it goes to your windpipe and lungs and can decrease this effect.
Nose-breathing has the added benefit of filtering out particulate matter.
bicycles.stackexchange.com/users/3363/djangodude
As far as I know, breathing through the nose might be the healthier option if you don't need the extra oxygen (for say, intense uphill efforts).
Breathing through the nose will optimize the temperature, humidity of the air you breath in and also makes any pollution reach your lungs a bit less probable.
There are also some other subtle effects, but since I'm no expert, I won't go into that.
You might want to research that if you are very interested.
During an intense effort, I would say deep (diaphragm) breathing through the nose and the mouth is the most effective way to fill your lungs with fresh air.
You might want to practice diaphragm breathing off the bike as well to get used to it.
It isn't very natural for the first time.
So the tldr version is: Nose when not intense effort Nose and mouth when intense effort Also try to keep your mouth humid when you breath through it (drink very small sips of water frequently).
A dry mouth is very unpleasant and also unhealthy (gums, throat).
On the cotton wool (it's pure speculation):
maybe something is put on them, probably to clean up the upper respiratory tract from mucus.
bicycles.stackexchange.com/users/6038/kovacs242
I think it kinda depends on what type of ride you are going for.
Generally, breathing from mouth provides more amount of oxygen compared to from nose.
So it comes down to this IMO (& Experience):
Nose:
You have to respire quickly to get enough oxygen to lungs
Mouth:
You can breath slowly but you can still get enough oxygen into lungs very quickly.
So, breath
depending on how you are riding .
If you are sprinting you will want to get as more oxygen as you can into the lungs, but if you are going on a casual ride, breathing from nose is enough and safe (from bugs etc).
bicycles.stackexchange.com/users/847/Starx
Some people get "Exercise-induced asthma" --
a.k..a. exercise-induced bronchoconstriction -- they say it's from breathing (air
that's cold, dry, and/or polluted) through your mouth.
It's especially true of/among cyclists (who exercise for a relatively long time).
If that's you, you might it's worth it to breath in only through your nose.
It may not be easy to tell whether you get (mild) asthma -- given that the symptom is, like, "shortness of breath while exercising".
I think it may be "unable to breathe as deeply as normal, either in or out" -- but don't take my word for it.
Youtube:
Why Do So Many Pro Cyclists Have Asthma?
bicycles.stackexchange.com/users/1097/ChrisW
I just watched the 'Road to Glory' documentary about Team Sky.  I noticed when they were warming up on their Turbos, some of them had cotton wool up their noses.  Is this because it's better to breathe through your mouth than nose?  If so, why don't they have cotton wool up their noses during the actual race?
Is it better to breathe through your nose or mouth when cycling?
bicycles.stackexchange.com
<training><technique><exercise>
bicycles.stackexchange.com/questions/12861
bicycles.stackexchange.com/users/4589/Steve Dunn
Most people agree that breathing through your nose is more benficial under normal conditions. It has the advantage of warming and moistening the air and filtering out dust and pollution. However, breathing the mouth provides more oxygen and thus is better when under extreme physical effort, for example climbing steep hills. Some athletes use cotton containing decongestent to free up their passages before a ride. Breathing through your nose warms the air and filters out some particles, which decreases the chance of irritation and asthma symptoms. Nose breathing provides less oxygen, but may be healthier and more suitable for warming up and less intense efforts. When sprinting, athletes breathe through their mouth and nose together.
Some athletes use cotton containing decongestent to free up their passages before a ride.  Breathing through your nose warms the air and filters out some particles, which decreases the chance of irritation and asthma symptoms. Nose breathing provides less oxygen, but may be healthier and more suitable for warming up and less intense efforts. When sprinting, athletes breathe through their mouth and nose together.