がんばれ is the imperative form of がんばる meaning “to stick to something despite difficulties”. The imperative がんばれ is often translated to “Do your best”, “Hang in there” or “Good luck”. It’s mainly used to encourage people to try hard or used before a performance to say good luck.
Japanese imperative verbs sound strong and aggressive so people normally tend not to use them but がんばれ has become a saying that’s commonly used. But if you want to sound softer, you can say がんばって or がんばってください in polite speech.
The response to it is usually うん、がんばる in casual speech, はい、がんばります in polite speech, and you can add ありがとう or ありがとうございます after that.
カカロット as you may know is Goku’s sayan name (Kakarot).
お前 means “you” in a manly way. Depending on the context, it can be rude but in this case it’s used in a friendly way between guys.
The が particle is used to mark the subject and emphasise it. So he is emphasising お前 as this is the first time that Vegeta admits Goku is stronger than him, so it’s like “it’s YOU, NOT ME”.
ナンバーワン is “number one”. As you know, words imported from foreign countries are written in Katakana, but the pronunciations are a bit different.
Katakana words mostly have equivalent Japanese words, so the equivalent Japanese for ナンバーワン is 一番, but katakana words are often used to make it sound cool or casual.
Lastly, だ is the auxiliary verb to declare and affirm the statement.