Learn German: Zuerst wasche ich die Tomate.

In German, the conjugated verb must always take the second position in a main clause. Because we started the sentence with the time adverb Zuerst (First) in position 1, the verb wasche must immediately follow in position 2. This rule pushes the subject ich to position 3.

Could I use Erstens here instead of Zuerst?

Not quite! Erstens means firstly and is mostly used when you are listing points in an argument or giving reasons. When you are talking about a sequence of actions over time (first I do this, then I do that), zuerst is the natural choice.

Is die Tomate in the nominative or accusative case here?

It is in the accusative case because the tomato is the direct object receiving the action of being washed. However, the feminine definite article die looks exactly the same in both nominative and accusative, which makes it easy to handle here!

I remember the verb waschen having an umlaut. Why doesn't wasche have one?