Mastering it Up
I've been asked on more than one occasion: "Should I get a masters degree?"
Short answer is no.
Long answer is that depends.
Computer Science Masters
A master in computer science will teach you more than a bachelors. You can look up any course list and see things about operating systems, networking, security, parallel computing, etc. All of these things are skills which could be good to have depending on your field of interest. But they are only skills, not experience. Compared with the BS in the same position, you will probably accelerate a little bit faster but will still face the same hurdles that the BS faces because of his lack of experience. Dealing with coworker's code, effective communication, developing software the company's way and dealing with ambiguity are all things you can't learn in a class.
What an MS will give you is experience in writing a thesis, dealing with professors, academic protocol and learning lots of nitty gritty details about a particular area. These kind of skills are great if your aim is to become a professor, industry expert or researcher. Developing ideas and researching them is more in the realm of science where developing software to solve a problem is in the realm of engineering.
It's not that an MS in CS won't benefit you when trying to solve engineering problems, it's that you will learn more about the industry by being in industry the extra year or two you would be spending earning your graduate degree.
Software Engineering Masters/Double Major
Getting a degree in Software Engineering on top of your Computer Science BS can help you in the long run because it teaches concepts that are hard to see at eye level. Many things in Computer Science can be learned from analyzing the code and taking a class on a particular subject. Software Engineering concepts can be things that are applied over an entire product cycle spanning many years. These are hard to recognize if you are not looking for them.
Good process is hard to learn and bad process is even harder to overcome without the foreknowledge of what good process is.
So again, is an MS in Software Engineering necessary? It's a choice you will need to make given your circumstances and progress. Even if you don't get a degree in Software Engineering, learning the how and the why of good software processes will help you in the long run.
MBA
An MBA is not necessary to a dev starting in industry. It can be useful later when you are a manager of manager of managers and your decisions begin to make or break the product in the market place. An MBA is something you can get later, whether at night classes or taking time off to finish it. Plus many large companies will give you tuition assistance.
Starting in industry also gives you the opportunity to see if the business end of software is what you really want. Many people start fresh from college thinking they want to become a Vice President and only later learning that they love the technical aspect of software development more than the managerial and business end. If this is you then the MBA and the time spent getting it kind of go to the way side. Get some experience and then decide which career path you want to take.
It is not always the case that the best way to get ahead is to become a manager, Microsoft has Technical Fellows whose "technical vision, expertise and world-class leadership is commensurate with that of a corporate vice president". There are opportunities not involving management, ask your recruiter about what the developer career path is.
But what about more pay!?
Some companies hire people with an MS at a higher pay rate. But they've been in school for the last 1-2 years, spending money and not making money. A BS who comes in at a lesser pay rate has been making money and has had those 1-2 years to get promoted and compensated up to or beyond the point that the MS will be. Plus you have that 1-2 years of seniority which can come into play big time when it's time to get new equipment. You'll have that juicy dual core and the new hire masters will have the lame single core. Join early, dual cores rule.