What is the difference between gambare and ganbare and ganbate ?Feel free to just provide example sentences.

がんばれ = ganbare = gambare (actual pronunciation)
がんばれ is usually written as ganbare because ん is supposed to be written as "n" in the most standard style of Romaji, though the actual pronunciation of ん in this word is "m". It's just a matter of which do you give priority to, a standard way of writing or an actual pronunciation.

がんばって = ganbatte = gambatte (actual pronunciation)
The same as above about "n" or "m", and this is a softer and more polite way to say がんばれ. There's no big difference between がんばれ and がんばって except that がんばれ sounds more powerful.