"I am drinking and I am eating."
Translation:Ich trinke und ich esse.
No, that's not it.
German words don't map to English words one-to-one; the grammar is different.
For example, German doesn't distinguish between present simple and present continuous the way English does, so that ich esse can correspond to either "I eat" or "I am eating"; also, German has a different verb ending for nearly every pronoun while English only has -s for "he, she, it" but no ending for any of the other ones.
So you have:
- ich esse
- du isst
- er isst; sie isst; es isst
- wir essen
- ihr esst
- sie essen; Sie essen
in the present tense, for "I - you (singular, informal) - he, she, it - we - you (plural, informal) - they, you (formal)".
So "we eat" would be wir essen but "I eat" would be ich esse.
And "I am eating" would also be ich esse but "we are eating" would also be wir essen.
Something like ich essen, wir esse, ich bin esse, ich bin essen are not possible.
I wrote "Ich trinke und esse." and Duolingo says it's correct. So can I omit the subject (like in Spanish) and say "Esse und trinke."?
No, you can't.
In this sentence, you can omit the second subject because it's the same as the first, much as in English "I eat and drink" or "I am eating and drinking", but you can't omit all subjects entirely in German the way you can in Spanish.