I feel like the mobile first is a ruse in some ways that investors love to hype. I had a startup a few years back (mid 2007- late 2010) that got some pretty good angel money. A pretty decent amount to get everything launched and keep us going and keep us running over the years. We rolled out a web site first and slowly built our paying user base up.
But as soon as the iPhone came out, my partner decided that we had to be a mobile company. He told me to remove the website, truncate our db, and start from a 'clean slate'. I was totally against it and years later, we never made anywhere near the money we had before we went mobile (company 'IP' was chopped up and sold off last year). But the thing is, our app really was location based, so from one point of view, it definitely made sense to have a very strong mobile presence. But we killed off our main revenue source to be a 'mobile company' - the stupidest thing ever and I'm a total business 101 fail.
Sure, mobile (as per Mary Meeker and others) has not reached its full potential. But if I were starting a new project, I would definitely consider if mobile first makes sense. I am primarily an iOS developer, but have built on many languages/platforms on the server side for years. I love making mobile products and I believe that there's pretty much always a place for it. But the whole 'mobile first' argument only makes sense for products that rely on the unique traits mobile offers.
Companies/apps like Uber, Wikipedia, News, Music stores, ticketing, Foursquare, etc, make most sense to go mobile first (even tho some may not have). They can be used away from the desktop easily and usually have a benefit in the real world. But unless you product fits into something like that, I would agree more with Vibhu and do mobile later.
And whatever you do, don't kill your main source of revenue to be a 'mobile company'!