The verb “cheer” and the phrasal verbs “cheer on” and “cheer up” may sound similar but they have very different meanings.
To “cheer” means a loud cry that shows approval or excitement.
“Everyone in the audience cheered loudly after the stellar play.”
“The baseball stadium was overwhelmed with the sound of cheering fans.”
To “cheer on” means to encourage someone by cheering (the verb we just talked about). “The crowd cheered on the final runner in the marathon.”
Again this involves loud cries, and is usually used at competitions. It usually happens in crowds, but a single person can do it too:
“The rest of the audience was silent, but Thomas stood up and loudly cheered on his daughter during the talent show.”
To “cheer someone on” with more mundane tasks is possible, but would be a strange social situation.
“Mary’s boss cheered her on as she worked” would have the boss loudly yelling something like “Hey you can do it!” to his employee.
“Cheer up” is a little different. It means to become happier, and is used when someone is sad.
“I was lonely my first week at uni, but a call from my mom cheered me up.”
“After his daughter lost the talent show Thomas bought her an ice cream to cheer her up.”
If you tell someone to “Cheer up” it means you think they are sad and are telling them to try being happy.
A: “I’m sad that we are moving to a different city.”
B: “Cheer up! You’ll make great new friends quickly.”