Don't use periods at the end of phrases or dependent clauses. If you do, you create sentence fragments. When she visits cities in the East, Tracey expects bad weather.

Do phrases have period?

Most don't have periods. The ones that do do so to be more assertive, which is not something your particular tagline has to convey. The only exception is Microsoft's which is two separate clauses.

Do you put periods after phrases?

Quotation marks and other punctuation marks

Sentence-ending punctuation is a whole different story. In the United States, the rule of thumb is that commas and periods always go inside the quotation marks, and colons and semicolons (dashes as well) go outside: “There was a storm last night,” Paul said.