First off, the more proper way to put it is "Permit to Carry" - In Minnesota, you can carry both concealed and open. Some states, you can only carry concealed.
Are you a felon? No? Well, then getting a permit is easy

First, you need to take a Permit to Carry class by a certified instructor. There are many instructors all over the state. The class runs $150-200. After completion of the course (which is usually on a Saturday or Sunday for 8 hours and then some shooting at a range), you will receive a certificate of completion from the instructor.
Now the course is done and you have your certificate. Head to your local Sheriff's Office and submit an application for the permit. They have within (30) days to either issue you the permit or deny your request. They cannot deny your request unless you have been convicted of a felony or are restricted from owning firearms. Minnesota is a "Shall Issue" state, meaning they will issue the permit to anyone who can carry it legally. Some states are "Will Issue", in which case, if they deem it unnecessary for you to carry a pistol, they will deny your request.
So... How hard is it? It isn't. Just get in contact with a Permit to Carry Instructor, take the 8 hour course, and within 30 days, you will be packin'

Get yourself a Glock (I'm biased), and a good Inside the Waistband Holster.
More information here:
http://www.dps.state.mn.us/bca/CJIS/Documents/CarryPermit/PermittoCarry.htmlIf you are in the Metro area, one of the best instructors in the twin cities is Joel Rosenberg. He wrote the book on carrying in Minnesota. His site can be viewed here, and it also includes upcoming courses:
http://twincitiescarry.comLet me know if this is helpful, and if you have any other questions, I'd be happy to answer them for you.
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So after typing all of that, I realize you are from Missouri. I am not as educated on non-resident carry, you're best to contact Joel at 612-501-8229 and he'd be happy to answer all of your questions pertaining to non-resident application and issuance.