Mobile first is touted by some really big names in our industry. Jeremy Keith, Brad Frost, and Mr Mobile First himself Luke Wroblewski has all repeatedly pushed for a mobile first website approach.
Strategically it is the way to go.
By starting mobile first with your responsive website you are able to make the tough decisions about the content. Mobile first often lends itself to a linear layout model, although this is a general truth and not necessarily a rule.
The great thing about this approach is it forces you into a content hierarchy. The most important piece of content is on top, followed by the next, followed by the next. Unlike a desktop approach where you can put two pieces of content side by side for equal footing, a mobile approach means one is above the other.
How to decide your order
There are many techniques, card sorting is one of the most common. I recently came across a great article by Brad Frost who laid out what I think is a great approach to a responsive project (not just the content considerations).