Brendan always said he only needed two things to be happy and healthy. friends and family.

Correct sentence- Brendan always said he only needed two things to be happy and healthy: friends and family.

Commas are used to separate a series of items, to separate adjectives, to help conjunctions join two independent clauses, to signal quoted material (along with quotation marks), and to set apart clauses and phrases otherwise. Colons do not function like commas, and are used to signal a list.

Because he had recently moved to a new city Peterson was sometimes lonely.

Correct sentence- Because he had recently moved to a new city, Peterson was sometimes lonely.

Commas are used at the end of an introductory adverb clause, in this case "Because he had recently moved to a new city." Adverb clauses are a group of words with their own subject and verb that together act like an adverb, describing a verb, adjective, or another adverb.

San Deigo is suffering from a major drought; umbrellas are a common sight in the rainy city of Chicago.

Correct sentence- San Deigo is suffering from a major drought. Umbrellas are a common sight in the rainy city of Chicago.

There are two independent but illogically related clauses in this sentence. Droughts in California have little to do with umbrellas in London. Because these ideas are unrelated, they cannot be joined with a semicolon.