"Det finns både te och kaffe."
Translation:There is both tea and coffee.
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I believe that since neither object is specifically quantifiable, you would go with "is" in this situation. Remove the word "both" and it becomes easier to see.
Examples: - There is [both] tea and coffee. - There are [both] cats and dogs. - There is [both] beer and wine. - There are [both] forks and knives.
Både is used with och as a conjunction, as you see here, to mean "both x and y."
Båda is used as a pronoun, and basically whenever you wouldn't want to say both without "both x and y," such as:
Två bilar i krock. Båda började brinna. (Two cars in crash, both caught fire.) Or
Jag skriver ut på båda sidorna av papperet. (I print on both sides of the paper.)