Learn German: Sie trinkt entweder Kaffee oder Tee. - She drinks either coffee or tea.

The pair entweder ... oder means "either ... or". Here, it simply connects two nouns (Kaffee and Tee). Because it is only linking the objects rather than whole clauses, the main sentence structure doesn't change. The verb trinkt stays in its normal second position.

Could I start the sentence with entweder instead?

Yes, you can! If you put entweder at the very beginning of a sentence, it usually takes up the first position. To keep the verb in the strict second position, the subject and verb must swap. It would become: Entweder trinkt sie Kaffee oder Tee.

Why are there no articles like "den" or "einen" before Kaffee and Tee?

German drops the article when talking about general, uncountable substances or liquids. Since she is drinking coffee or tea as a general category rather than one specific cup, we leave the articles out. This works exactly like it does in English.

What grammatical case are Kaffee and Tee in here?

They are both in the accusative case because they are the direct objects of the verb trinken (they are the things being drunk). You can't see the case here because there are no articles, but if you wanted to say "a coffee or a tea", you would use the accusative endings: entweder einen Kaffee oder einen Tee.