We can't sign any new agreement with the EU until our old ones are ended, which won't happen instantly.
We can begin thinking about new agreements with non-EU couintries as soon as we want to, but they may very likely want to see what arrangements we come to with the EU before they jump into finalising the horse-trading we can offer to them, and what they are prepared to offer to us.
Trade agreements are not just about what tariffs will be imposed in either direction, they are about what size of market we can open up to a trading partner (a lot smaller once we are out of the EU - unless we can swing something good) in proportion to the size of market they can open up to us, and how much we need the deal against how much THEY need the deal.
We might find ourselves (as Switzerland did with one EU deal) being told "Yes, if you will let us sell in your country right away, we will let you sell in ours from 2020"