Are is used for plural subjects, whereas am is used for singular subjects. Jim, John, and I is a plural subject (3 people), so the correct form is "Jim, John, and I are going somewhere."
There's some nuance to this rule. You might say "The king and I are wonderful friends," when you're talking about your lofty connections, but you'd say "The King and I is a wonderful musical", since here The King and I is the title of a play. With some subjects you can use either, depending on whether you want to evoke plurality or unity. For example, you can say "The Beatles is my favorite band" when referring to The Beatles as a group, or "The Beatles are all excellent musicians", if you prefer to evoke a group of individuals. You can see that in this case the plural is more popular, but the singular form still enjoys regular use.
Rarely, writers will even go so far as to use the plural verb for singular nouns, when they want to emphasize the individuals rather than the group. "The couple are separating" is a common example. Likewise, the band are tearing through their set is better at evoking individual musicians than the singular form.