There are two paths, terminal MS and PhD.
Terminal MS deepens your knowledge, and may or may not be thesis based. For most specialties it isn’t necessary to get a job, but with a Masters you will probably stand out in a hiring field and get a pay bump.
PhD is longer and should result in research that adds to the knowledge base of the field. It’s useful for very cutting edge things in industry or can be parlayed into a university career. It’s a major commitment though, probably 3 years at a minimum and 5 or 6 at the longest. It really depends on how your research progresses.
Where you do your graduate work depends on a multitude of factors. Many terminal MS programs are unfunded and course based only. This means you have to pay, so cost and relative return on investment can be a factor in choosing a program. A few are funded and thesis based.
In choosing a PhD program, you find the advisor you want to work under, regardless of the school they’re at. So, let’s say you’re into high speed aerodynamics. Texas A&M might be better than either MIT or GT. You’re picking a person, not a school.
https://www.science.org/content/article/should-you-go-grad-school