Learn Danish: Jeg køber en agurk og tre tomater. - I am buying a cucumber and three tomatoes.

Danish does not have a separate present continuous tense (the '-ing' form in English). The present tense verb køber covers both actions happening in general and actions happening right now, so it can mean either 'buy' or 'am buying' depending on the context.

Is there a rule for why it is en agurk instead of et agurk?

There is no strict rule to guess this! Agurk is a common gender noun (an en word), just like about 75% of all Danish nouns. Because you cannot usually guess a noun's gender just by looking at it, it is always best to memorize the article en or et when you learn a new word.

I see tomater ends in -er. Is that the standard way to make a word plural?

It is one of the most common ways! The singular is en tomat. While many words take -er, others take -e or have no ending at all in the plural. You will start to spot patterns, but plural endings still need to be memorized.

How would the end of this sentence change if I wanted to say 'the tomatoes' instead of 'three tomatoes'?

You would need to use the definite plural form. In Danish, you make a plural noun definite by adding -ne to the end of it. So, tomater becomes tomaterne, and you would say: ...og tomaterne.