Well I'm not talking about putting this off forever, but it would be prudent to decide who is going to lead the negotiations with the EU over the terms of the UK exit, it would also be a good idea to decide what kind of deal we would want, do we want EU-lite, a complete separation, something in between.
If DC has said that there will be a new man in #10 by October then it's likely that they will invoke article 50 before the end of November.
~20 weeks from polling day to starting the formal procedures isn't that long, in the general scheme of things.
If the EU imposes punitive measures on the UK to "make an example" then it'll end up hurting them too. We aren't leaving Europe. The UK isn't going to migrate to the mid Atlantic. We still need to trade with European countries, and they will benefit from trading with us. UK citizens will want to go and work in Europe and Europeans will want to come and work and live here.
e.g. German cars. We buy lots of German cars in the UK. (they make damn good cars) About 1/5th of German car production is bought by the UK (~820 000 cars last year) Germany are unlikely to want to put a big dent in that figure by imposing lots of tariffs. The UK certainly won't want high EU tariffs so it's highly likely that a post Brexit deal will include them.