Speaking from experience, so I won't sugarcoat this...there are hundreds and hundreds of recent grads who move to DC every year with your exact goals in mind. All jobs (not just DoS, DoD, DHS, etc.) in foreign policy work are extremely competitive, even landing a job with a decent policy think tank will require some combination of a masters degree, proficiency in 2-3 foreign languages, embassy experience, a security clearance (this might be the most difficult), government experience, and of course the hallmark of D.C., who you know.

I don't want to stamp out your dreams but you need to understand that foreign affairs has extremely high barriers to entry and is easily the most 'blue blooded' policy sector in the city. A law degree should not be your priority if this is your goal.

One of the best ways you can get into this field is finding work at an embassy as an intern. Embassy experience is extremely valuable as you will more than likely be given security clearance just to work there, also you will have a great chance to learn a new language, and gain gov't experience. Finding one of these internships is not easy but it is certainly possible.

From there I would try to go get your MA at Georgetown, GW, or Johns Hopkins SAIS (School for Advanced International Studies). If you can get into one of those programs you'll be given a great opportunity to find the sort of entry level jobs that will help with the who you know aspect. Hopefully from there you can meet someone who will help take you to the next step and help you find work as a full time analyst.

Working in foreign policy in DC is not the sort of decision you make on whim, it is a life-long career that takes a lot dedication, commitment, time and patience.