Learn Danish: Jeg besøger ham ofte.

Han means "he" (the subject doing the action), while ham means "him" (the object receiving the action). Since "I" am doing the visiting in this sentence, we use the object pronoun ham.

Could I put ofte before the verb, like in English ("I often visit him")?

No, Danish word order is strict here. In a normal main sentence, the verb must sit in the second position (this is called the V2 rule). Short adverbs like ofte (often) generally come after the verb and the object pronoun, making Jeg besøger ham ofte the correct order.

Why does ham sometimes sound like "am" when native speakers say this sentence quickly?

In everyday spoken Danish, the "h" in the object pronouns ham (him) and hende (her) is very often dropped. So instead of a crisp "h" sound, besøger ham will often blur together to sound like "besøger am".

How do I know if besøger means "visit" or "am visiting"?

Danish does not have a separate continuous tense like English does. The single present tense verb besøger covers both "visit" and "am visiting". You can tell which one makes sense based on the context of the conversation.