For me - Britain made 3 errors:
1. Holding a referendum on such a complex issue. Leaving the EU is impossibly complex. The average MP would have no idea about the complexities of leaving the EU - both in terms of actually unwinding the relationship, and in forging new relationships with the EU, and the rest of the world. These are not issues that are best decided by the voting public - and certainly not when the vote is really about broad emotionally-charged issues, and the campaigns are boiled down to tiny sound-bites.
I think it is fair to say, there are carious aspects of the EU that are unappealing to the average Britain. And, I think those are fair observations. But, to boil down those issues into a simple Leave or Remain option was a mistake, and a big mistake to make it a binding referendum. No voter could possibly have understood what "Leave" would look like after negotiations - but they all were seemingly unrealistic in their expectations of how the EU would treat them in the negotiations.
2. The second mistake Britain made was multi-fold. Once the referendum was complete - there was a slight lean towards leaving the EU. It was not a mandate - by any stretch of the imagination. But, that referendum did give Britain a bit of leverage that it did not ordinarily have in its relationship with the EU. Instead of jumping straight into Article 50, Britain would have been better served using that result to negotiate with the EU over various important issues - such as immigration, and local sovereignty. The EU would presumably not have given Britain everything they wanted - but I suspect under threat of leaving, Britain never would have had as much leverage to gain some concessions. And compromise is at the heart of any negotiation.
3. The third, and perhaps most glaring, error Britain made was in the current government not backing May's negotiated settlement. I think many in Britain were expecting the EU to roll over and give Britain everything she wanted in the Brexit negotiations. That was never going to happen. The EU effectively hold all the cards here - they have no incentive to give anything away to the country who is leaving. The British government should have understood that up front, and when they authorized May to negotiate the terms, they should have done so, knowing they will not get everything they want. Once the negotiating team was empowered - the government should have accepted their best efforts and gotten on with the business of forging new agreements with the US and Asia - and then worked, over time, to mend the fences with Europe, and delivered better trade agreements.