I am often asked what the oldest continuously published American magazine is. The answer is the venerable American Journal of Science, originally published by Benjamin Silliman. If you go to Yale today, you can go into Silliman Hall. The current issue is Volume 309 Number 2 and AJS is still the pre-eminent Earth Science journal published in America.
The New England Journal of Medicine actually goes back to 1812, but it went through a number of name changes so AJS still gets the prize for the longest continually published title.
Of today's literary magazines, the oldest are Harper's Monthly (1850) and Atlantic (1857).