Here's something I just realized I didn't know how to say in Japanese. What is the Japanese equivalent for words and set phrases used to express excitement for what someone is doing, while also cheering them on and actively encouraging them to keep it up?
Common ones heard during many types of games are:
People often say these repeatedly ("行け行け行け！"). Other verbs are also used when appropriate ("押せ押せ押せ！", "走れ走れ！").
in conjunction with a noun is common.
(As @naruto mentioned you can string this together like in English. You don't need to specify who at this point as it can be contextually obvious.)
An example of this in use would be in Pokemon, where Satoshi ("Ash") will say
行け！ ピカチュウ！ ("Go, Pikachu!")
You can use 頑張れ for many other situations, however. (e.g. 面接頑張れ！ but the meaning would totally change into a rather shocking phrase if you were to say 面接行け！)
for most purposes for cheering someone on (well, 励ましの言葉) are pretty similar, but depending on your use case you may want to pay attention to what types of people use them in what situations and see if that fits.