From what's been said so far by senior politicians, diplomats, and trade economists, the UK's priorities will start with a new arm's length trade agreement with the EU based heavily on CETA.
Parallel with that will be agreements duplicating the existing terms of EU trade treaties with third parties. The UK has already been conducting negotiations with those countries, and the emphasis is on trade continuity along existing terms. Finalising these may be delayed until enough detail has been worked out with the UK-EU negotiations to ensure the third party treaties don't conflict with that one.
Beyond trade continuity, the UK can look at joining existing multi-lateral trade agreements as these would allow joining a system that is already stood up and working, as opposed to one that had to be started from scratch. In the past couple of days I posted a story on one of the Brexit threads where a trade economist recommended the UK joining the CPTPP (Australia, Brunei, Canada, Chile, Japan, Malaysia, Mexico, New Zealand, Peru, Singapore and Vietnam). Other parties have said this would be a good idea as well. To put CPTPP into perspective, the total GDP of CPTPP countries with the UK added would be about 90% of the size of the total GDP of the EU without the UK. It's not a tightly knit group, but the size means it's worthwhile considering.