If you can handle it financially, the startup scene is really a god-send for a lot of new graduates in STEM fields. They're less likely to take you to task over your experience, because your lack of experience makes you cheap, and cheap is the bottom line in most startups. They can offer a great opportunity to grow and advance quickly because you'll likely be asked to do things that are beyond your initial job description. You can really feel invested in the work you're doing because odds are you'll be offered some equity or possible stock options depending on how far along they are. Plus there tends to be a lot of turnover, which can really suck on one hand, but also helps you build a really solid network very quickly. "Oh yeah. I worked with John at <company_X> and he's awesome." goes WAY beyond anything your resume will ultimately do for you. I hire for positions a couple times a month and if someone in their network is in my network and has a good relationship they immediately go to the top of the list.
The problem of course is equity doesn't make student loan payments, and stock options that vest after 24 months of employment don't put food on the table. So depending on how you are in terms of debt and the like, you might be better off trying to start off on the low rung of a larger company. Yes, you'll probably end up spending 95% of your work day on crap that you dislike, but you're building your own brand equity by attaching your name to a company people recognize. I have 10 years of experience with web development and surrounding disciplines with 4 different companies. In all of that, the first thing I'm asked at a job interview is "tell me about the time you spent at <insert major tech company here>." 6 months. They care more about the 6 months I had with that company than the other 9 1/2 years in the industry.
Also, I echo the sentiments about holding off on your MBA. It's not going to give you a notable jump over your competition. I know a number of people who waited to go back for their MBA and for the most part they're doing much better than the people who got it before entering the work force for a few years.
Lastly, but perhaps most importantly, be humble. You need a job, so realize that any job is an opportunity. If you get an interview, learn about the company and ask lots of questions about the position while you're talking to them. Networking isn't just the people you know, it's the people
they know. If you're #2 on the list at Company A, the hiring manager might get you to #1 at Company B because they know someone over there. Every time you're in a room with 3 people, or on a call with 1 person, you're making an impression that will spread to exponentially more people without you even realizing it.
Good luck.