All students at the four(!!!) undergraduate colleges are Columbia University students, and all Barnard professors are also Columbia University faculty.
Students from the various colleges take classes on either campus, whichever fits their schedule best, or where a course happens to be taught that semester.
Neither professors (nor fellow students) care, which particular college you are enrolled at.
That also goes for other resources: As a first-year student, my daughter got a Columbia University fellowship towards summer studies in Venice, that included upper class people, and even grad students!
Yes - my daughter took/picked classes on both campuses, based on various criteria. The choice of course location never impacted her direct interaction with her professors.
She was very conscious about doing “her part”, and then making full use of office hours, and TA resources (in larger classes) and, in all her years there never expressed concerns about accessibility, responsiveness or lack of hands-on support (quite opposite - she often mentioned how helpful faculty had been.)
By the time she had decided on a major, professors suggested internship/practicum opportunities (even at other selective Manhattan universities), as well as sharing first-hand insights into different professional paths.
As far as your specific first-year research opportunities - that clearly is dependent on your field, on the current needs of professors vs. what you bring to the table.