Learn Spanish (Latin America): Compro un regalo y un libro.

Both regalo (gift) and libro (book) are masculine nouns. When you want to say 'a' or 'an' before a masculine noun, you use un. The word uno means the number 'one' when it stands alone, but it drops the -o when placed directly before a masculine noun.

Why isn't the word for 'I' (yo) in this sentence?

In Spanish, the ending of the verb tells you who is doing the action. The -o at the end of compro already means 'I buy'. You can add yo at the beginning (Yo compro...), but it is usually left out unless you want to emphasize that you are the one buying it.

Can I skip the second un and just say Compro un regalo y libro?

It is much more natural in Spanish to repeat the article for each noun you list. So, even though English speakers might say 'a gift and book', Spanish prefers to keep both: un regalo y un libro.

Does compro mean 'I buy' or 'I am buying'?

It can mean both! In Spanish, simple present tense verbs like compro can translate to 'I buy', 'I do buy', or even 'I am buying' depending on the context of the conversation.