What is the best way to secure an internship, or scouting job, with an MLB team out of college?

As a former MLB baseball department executive, I have a couple recommendations for you.

Regarding an internship:
I would highly recommend attending the Baseball Winter Meetings held each December. The location changes each year. At the Winter Meetings, there is a program called Job Seekers Seminar. It's a program whereby baseball teams (mostly minor league teams) post job openings and where candidates have opportunities to submit their resumes and talk directly with the teams posting those jobs. It's a great avenue for young people to look for internships and/or entry level jobs in baseball.
Also, continue to search the website: teamworkonline.com where teams post jobs and oftentimes internships.

Regarding Scouting:
Scouting jobs for students right out of college are very scarce and there is no easy path. Each major league team may have 25 or so scouts who scout colleges and high schools in an effort to prepare for the June draft and another handful (less than 10) of scouts who are called Pro Scouts and scout the current professional players in the Major and Minor Leagues.
My recommendation is to find a full-time scout in your area and see if you can shadow him for a few games and potentially work for him or his organization as a bird-dog scout. Bird dog scouts may or may not be paid and attend baseball games that the full-time scout cannot get to. The bird dog scout is an extra set of eyes (and ears) for the fulltime scout and may be able to recommend that the fulltime scout take a look at a player you may think has potential minor league skills. If you attend a minor league game and/or college game you will see the scouts sitting in the sections behind home plate with radar guns and notepads.
Getting experience in watching and evaluating players is key even if it's in a bird dog position. Then hopefully, keeping your ear to the ground you will hear of a full-time or even paid part-time position with an organization.