difference between cheer for, cheer on, and cheer up

cheer - (хөгжөөн дэмжих, уухайлан дэмжих)"I cheered for my team" means you were happy about something your team did, so you shouted happily.
1 [intransitive, transitive] to shout loudly, to show support or praise for someone, or to give them encouragement
       ---- We all cheered as the team came on to the field.
cheer on - (хөгжөөн дэмжих, уухайлан дэмжих)I think "cheer for" and "cheer on" are pretty similar in meaning. You can cheer for your favorite team, or you can cheer them on
cheer someone ↔ on
to give shouts of encouragement to someone in a race, competition, etc.
       ---- The spectators cheered the runners on.
cheer up - (хөгжөөх, хөгжилтэй болгох, урам оруулах) If you wanted your friend to feel better, you would try to cheer him/her up. So, "cheer up"
cheer up and cheer someone/something ↔ up
to become more cheerful; to make someone or something more cheerful
       ----- Oh, come on—cheer up!
        ----- Give Mary a call; she needs some cheering up.

difference between cheer for, cheer on and cheer up

• 1. If you wanted your friend to feel better, you would try to cheer him/her up. So, "cheer up".
  
  2. You "cheer for" your favorite team.
  
  3. Looks like they are "cheering for" something.
  
  
  I think "cheer for" and "cheer on" are pretty similar in meaning. Number 2 and number 3 could probably also have used "cheer on".

• If my friend was having a bad day, I might tell her to cheer up. "Cheer" can also mean "happiness," and "up" can mean "increase," so "cheer up" can also mean "increase happiness". (You'd never say "increase happiness," because it sounds very strange, but it has that meaning.)
  
  You can cheer for your favorite team, or you can cheer them on. The two are very similar. Saying "I cheered on my team" means you were trying to encourage them. "I cheered for my team" means you were happy about something your team did, so you shouted happily.
  
  "My team was losing, so I cheered them on."
  
  "I cheered for my team when they scored a point."
  
  I hope this helps!

• I think the definitions above are good. There is a subtle difference between "cheer for" and "cheer on," but I think SpencerDub explained it very well. "cheer for" can also mean "root for" or "want to win" or something like that　（贔屓）. So if you are at lunch with a friend and they ask you about baseball, you can say "I cheer for the Giants" to show that you like them (even though you aren't "cheering" right then).