You can usually work as an engineer with an undergraduate engineering degree, like my sister-in-law does, and then sometimes even your company will help pay if you want a grad degree. To get into the grad engineering programs you take the GRE, just like you do for most other graduate programs in anything. There are subject GREs but in my experience most people don't even use them, just the regular GRE.
you can also work toward being a professional engineer (PE). first you take the engineer in training (EIT) exam, now called the fundamentals of engineering, around your junior year. then work under a licensed PE for 5 years and take the PE exam. most of the time PE's work in civil engineering or work as consultants. you don't need a PE license to work as an engineer
hmm I'm pretty sure my sis took the PE exam her senior year at Columbia engineering, and now she's a liscenced PE, so I guess you can take that whenever you're ready? She is a civil engineer. But in any case you can definitely work without grad school so it's a totally different ballgame than medicine