Yes cold viruses can be spread through breathing and every time you breathe out you spread the cold virus into the air which then gets picked up by other people when they breathe that same air in.
Colds are usually most common during the cold winter weather when people are cooped up indoors around people and when someone gets sick with the cold virus they are constantly breathing and spreading those cold viruses through the air.
If you go inside someone else's house and they have a cold virus or flu virus then you're likely to get infected with it as well unless you have a flu shot which can lessen your chances of getting the flu and possibly the cold virus as well.
Getting a flu shot before flu and cold season can help some but it can't guarantee you'll never get the flu or cold virus.
Even when you have the cold or flu virus you should try to get outside to get some fresh air and get everyone else outside as well as the fresh air is good for your health when you have the cold or flu.
I find that if I even just go outside on my patio and sit outside even during cold weather I get over the cold and flu virus much faster than I would when staying indoors all the time.