That was not necessarily a mistake. As a fresh college grad (and particularly as an engineer) I'd argue that you are ill-equipped to judge a good startup opportunity from a bad one. There is absolutely nothing wrong with spending time at a large company, benefitting from their experience in graduate intake, learning a lot of good, sensible practises, then applying them to a startup.
I've also recently become deeply skeptical of working for a startup as anything other than a founder or a very early employee. You don't benefit monetarily (low pay, bad stock options) and the work environment is frequently toxic and incredibly demanding. The only difference is you'll be making your bosses rich instead of making stockholders rich.
However, I'm at a more established firm now, and the fact that I can spend time building systems properly and work on self-development is invaluable. I think working for a big company, even a giant bureaucracy, out of school, is not a bad thing at all, provided you can spend a few hours independently working on side projects.
My first job out of college was at a large company and I worked with some old timers who had been in the industry for years and on code that, at least parts of it, had been written almost 20 years ago. Learning from my colleague's experiences and through the code was incredible and I am certainly better for it. These sort of experiences can rarely be had working at a startup.
Lately, I've noticed a meme from investors about discerning founders who can start companies and those that can grow beyond 250+ employees. The generally consensus is that there are very few entrepreneurs (Zuck for example being one of them) who possess the capability to scale. I'm not talking about scaling servers, I'm talking about scaling organizations as quickly as their revenues grow. Looking at it through the lens of an external person, this may seem trivial at first, but having been in the professional world for 8+ years now, there are some key challenges all companies hit when they hit certain growth (for example, hitting 50 and 250 employee milestones). Working for a big company can (notice I didn't say will) prepare you for that.
So, pretty much any business? There are so few companies out there, especially venture backed ones, that the employee headcount hasn't scaled with the growth. We just hear of the Instagram/WhatsApp acquisitions with low head count so often here in the echo chamber that we assume it's norm. To take that further, how many companies are at FB/SalesForce/WorkDay/Twitter-level that don't have over 500 employees (much less 1,000s)? I can only think of very few - 37signals, Valve, etc?
My larger point - VCs in SV want to fund billion dollar businesses and founders who can create billion dollar businesses. These billion dollar businesses need large amounts of headcount and a CEO who can take them there.
I also feel the same in regards to working for a startup when you're not a founder or a very early employee. It is completely true you likely will be making your bosses rich and not as much yourself, unless you're like an absolute rockstar their equity is still going to be at least 20-50x more than you (this is just a rough guess). If you think you can do it on your own, might as well.
With regard to this comment: there is nothing wrong with "spending time at a large company, benefitting from their experience in graduate intake, learning a lot of good, sensible practises, then applying them to a startup."
That is true. But there is also nothing wrong spending time at a small startup, learning good sensible practices, learning from smart people, learning from making mistakes, THEN applying these skills at large companies.
People often say "You'll learn a ton at amazon, then you can work anywhere." That is true, but you can also learn a lot if you work for a great startup.
The idea that you learn more at big companies is planted in students brains at these career fairs. And when GS and BofA come to campus to recruit. The idea spreads because startups don't come to campus and preach the reverse.
There are small startups; no one knows whether these will succeed, even if they're YC startups.
There are big(ger) startups that are clearly on the way up. Pinterest. Dropbox. Airbnb. Uber. Snapchat.
Then there are medium sized but fast growing startups. Quora. Stripe. etc.
The latter two categories are obvious choices.