Ok, so when people say the columbia core they usually mean the columbia college core, there is a seas core also, which has many aspects of the college core, but varies a little from major to major, like most seas majors need to take an intro comp sci class except chemical engineering majors, but basically you will have to take the above classes, plus:
1. gateway lab (1 semester)
2. a year of contemporary civilizations or lit hum or major cultures, ( i think this means two semesters in a row usually sophmore year)
3.either art hum or music hum - 1 sem
4. another semester of PE
5. a comp sci class - 1 sem
6. a preprofessional course, in any of the engineering disciplines - 1 sem
no frontiers of science, it's quite meaningless for seas kids (and for many cc kids also

)
i might be missing some stuff out
and of course you have to take classes in freshman and sophomore year that go towards your major (like mech E majors have to take a physics sequence) it will all become very clear once you get to columbia, or check out the bulletin, either way dont sweat it.