I am about to graduate high school and enter into an undergraduate program with a small, private university in the midwest with a vocal performance major. I was wondering if I completed the program with this university, I could go obtain a master’s with a more specialized university/conservatory in New York. The university I plan to get my bachelor’s from has a very good vocal program. I was wondering if the master’s in a city like NYC would really be beneficial to my career as a performer. If anyone has done something similar, or knows someone who has, I’d appreciate the input. Thanks! 
Short answer: Yes. I’d say that, the more experience you get with vocal performance anywhere (but especially in NYC), the better your voice gets and the more experience you have taking on different roles. Take part in different productions and work your way up to lead roles, no matter what.
As for graduate programs, I know Juilliard has a Master of Music. I also know that Yale has a great Master’s program for drama. What kind of vocal performance are you planning on going into? Classical? Broadway? Pop?
While @edgrad1981 isn’t wrong, I worry about the cost. Do you have funding for another program? In nyc?
If you’re looking for an inexpensive program (relatively) consider CUNY queens Aaron Copland school of music. They may have a program that you like. You can get in-state residency by living in the City for a year (working during that time).
@edgrad1981 Ideally, eventually, I’d love to go into musical theatre. I will be classically trained at the university I’m planning to get my bachelor’s from.
@Dustyfeathers I’m not extraordinarily worried about the cost right now. Most of my tuition for my undergrad program will be covered through scholarships, grants, etc.
I’ll look into that! Thank you very much