Learn Danish: Hun køber en lille trøje til barnet. - She is buying a small sweater for the child.

Danish doesn't have a separate "continuous" tense like English does with "is buying" or "are doing." The simple present tense køber covers both "buys" and "is buying."

Does the adjective lille change if the noun is an et word instead of an en word?

No, and that makes it special! Most Danish adjectives add a -t when describing an et word (a neuter noun), but lille (small) is an exception. You use lille for both singular en words and et words. Just remember that for plural nouns, it changes completely to små.

Where is the word for "the" in barnet (the child)?

In Danish, "the" is normally attached to the end of the noun rather than being a separate word in front of it. Because barn (child) is an et word, you add -et to the end of it to make barnet (the child).

Why does til translate to "for" in this sentence?

Prepositions rarely match up perfectly between languages. While til very often means "to" (like traveling to a city), it is also the standard word to use for "for" when you are giving something to a recipient, like buying a sweater for a child.