Rule one
If your sentence begins with an introductory element, put a comma after it. Even if it’s a short element, put a comma after it. In time, you’ll be putting this comma in without having to think about it.

Rule five
Complete sentences that are joined without a coordinating conjunction need a semi-colon instead of a comma; the semi-colon shows the end of one sentence and the beginning of the next. Semi-colons are often followed by a connecting word or phrase; however, a connecting word or phrase is not necessary. Sentences joined with only a comma are called comma splices; they’re among the most common errors that come up in college writing.

(Note: In the next-to-last sentence in the previous paragraph, there’s a comma after however because it’s an introductory element in the second sentence.)