I believe this is the timeline:
Article 50 gets invoked on March 31st 2017
We then start negotiating our withdrawal with the EU, how we unpick ourselves legally from the union.
On March 31st 2019 we leave the EU (unless all 28 countries agree to delay it, or the ECJ rules that the article wasn't actually invoked correctly)
On April 1st 2019 we have no trade agreements with anyone in the world, and can begin signing them. Our main trading partners, thanks to a fluke of geography, are those nearest to us, and that's Europe. WTO prices apply, and large tarrifs will apply until we sign a deal.
When Greenland left the EC (on it's independence from Denmark), it took them 3 years to agree a deal with the EC on a single good -- fish -- which accounts for over 95% of it's exports. There are somewhere in the region of 4000 categories of goods in international trade.
Of course Greenland is still ruled by Brussels. At least time time they can legitimately say they have no say in it.
http://www.itv.com/news/2016-05-10/what-lessons-can-the-uk-learn-from-greenland-leaving-the-eu/
But still, fun times ahead. Red White and Blue Brexit means Brexit. How do you like your Brexit in the morning? I like mine with a kiss.