I have a Bachelor's of Computer Science and about a year and a half of experience in programming. I'd like to be able to work in different countries and explore the world while I work. Are there embassy jobs that someone with my background could get into? If not, would getting a Master's degree in International Relations or something similar help? Would there be a job that combines the skills of the two degrees? I have dual Canadian-US citizenship, but since I've lived in Canada all my life I'm not sure I'd be qualified to work in an American embassy.
Probably not. In my experience with US Embassies it's only absolutely essential staff, so only people who talk face-to-face with customers and the public officials.
Something you could go for is an IT job, that requires constant travel to maintain and setup infrastructure at embassies, consulates, and diplomatic outposts. Unfortunately, it doesn't seem like you would like that, and usually these require security clearances and similar experience.
If you're only looking for the travel aspect, I'd recommend being a programmer with a company with large international outreach. If the company has some sort of software product that people from all over the world will buy, then you might be required to travel constantly.
My final piece of advice is to look into Incident Response in the cybersecurity field. Companies will pay other firms who employ these analysts to have them come to their site when a breach occurs to secure it. Sometimes this could go up to 75% travel. Usually you won't get right into it, but you could work your way up into that job.
There are certainly Foreign Service Officer positions (US State Dept representatives at foreign embassies) with a technology focus. Their listing of Specialist Career Tracks includes an Information Technology category.
Quoting from that page:
[Information Management Specialists] manage and operate worldwide information technology infrastructure.
With that said, there may be a trade-off between the technical interesting-ness of the work; but certainly the State Department probably has some relatively interesting IT problems.
I think a much better idea for what you're looking for is a
100% remote position. I've definitely seen those on job boards and there are even dedicated job boards for those. Wouldn't you rather have a job where they don't care where you are, so you go wherever you want, when you want, instead of one where they tell you when and where to go?