Colon: "here it comes"
Use a colon after a complete sentence to introduce a list, an explanation, an example or a quotation: She had one goal: to finish before dawn. The part before the colon must be a full sentence, so "The ingredients are: flour, eggs" is wrong; drop the colon.
Semicolon: "and here is a related thought"
She had one goal; he had none. Both halves are full sentences and neither explains the other; they sit side by side.
A quick test
Could you put "namely" or "that is" in the gap? Use a colon. Could you put "and" in the gap? Use a semicolon.
Capitals after a colon
In American style (Chicago, APA), capitalise after a colon only if what follows is a full sentence (APA) or more than one sentence (Chicago). British style generally uses lower case.
Frequently asked
Can a colon come after "such as" or "including"?
No. Those words already introduce the list, so no colon is needed.
Can a colon join two sentences?
Yes, when the second explains or expands the first: "I knew why he was late: the roads were flooded."