'Wherever I go'?
If I were to write 어디든지 가도 맨날 우산을 가져와요, would it be correct?

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Bună ziua!

Wherever I go 
어디(를) 가더라도

어디 가더라도 항상 우산을 가지고 다녀요. is more natural

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어디든지 가도 항강 우산을 가져갑니다.

^Luke is correct above. Your movement verbs should stay consistent!

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I'm actually not terrible confident about this but I think I'd say

어디에 가도 항상 우산을 가지고 갑니다

I think your movement verbs should be consistent between the two clauses. it's strange to be going in the first clause and coming in the second