"I went on a date with my boyfriend on the weekend."
kare can either mean he or boyfriend (very context sensitive). kareshi is (mostly) explicitly boyfriend.
(A person talking to you and gesturing at the man besides you)あなたのかれですか。Is that your boyfriend? (のかれ made it more obvious that it's meant as boyfriend, like (あなた/わたし)のかのじょ for your/my girlfriend.
The "shimashita" form of "suru" is the verb here. The verb "suru" usually turns a noun into an action and is preceeded by an article like (wo) or (ni) in some other cases.
Making a compound of a foreign word (kiss, date, knock, etc) with a form of "suru" is rather common in contemporary Japanese.
The verb can be a bit complex in some cases and it's used quite a lot, so it's worth reading a more detailed explanation.
週末には彼氏とデートしました。was marked wrong. Am I misunderstanding the function of には here?