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The problem with arriving at a negotiating position was that the Conservative Party has never had a single view of what was wrong with the UK's position within the EU. There are factions ranging from "It's just fine as it is" to "Hard Brexit immediately."

For many of the Conservatives in favour of exit, their position was about the leaving, not the end result, for which they didn't have a plan. They weren't expecting to win the referendum, and weren't prepared for it. Some of them may not have really wanted to win, since campaigning is much easier than implementation.

This meant that arriving at a negotiating position involved a large number of compromises, taking time and producing a very vague document. There was a lot of pressure to get the process started from the "Leave now!" wing of the party, which appears to have been responsible for the timing of the triggering of Article 50, although we won't know for sure until May publishes her memoirs.

The public assumption was that the delay had allowed getting a negotiating position ready, but that was apparently false, given how much longer it took before a position was settled. Worse, the position was not backed by large sections of the party, producing the current factionalism, and lack of ability to get the current deal through Parliament. Calling of a General Election and failing to get a large Conservative majority must have mucked things up too.

The "Hard Brexit Now!" group are the only people who have a robust plan, as of December 2018, which is to prevent anything else happening until the UK leaves (on WTO terms) by default. It is actually within their power to do this, in the current Parliamentary situation. The Prime Minister can't get her plan through without co-operation from the Labour Party, which seems highly unlikely at present.

The group in favour of a second referrendum have a plan, but they don't have the ability to make it happen. The Prime Minister is in the same position, with a plan, but no ability to execute it.

Negotiating for Brexit was the job of the executive wing of the UK government, which means the Cabinet, led by the Prime Minister.

In February 2017, shortly before triggering Article 50, the government published the UK negotiating policy for Brexit, setting out what it wanted. However during negotiations it became clear that this position was unrealistic, and hence the Cabinet had to agree a revised position in the Chequers Plan.

However the government still got very little of what it wanted, even from the Chequers Plan. The argument within the UK now is about whether what it got is better than what it has now, or what it would have if it left without any agreement.

Theresa May knows that she needs something more to get her deal through Parliament, but there is no consensus in the Government on what one or two things it should push for, and what it might be prepared to give up in return. This leads to the vagueness that the EU is complaining about.

Article 50 was triggered in the absence of a consensus within the party because the U.K. needed "to allow withdrawal negotiations to begin".

Wikipedia says this:

On 29 March 2017 the United Kingdom (UK) invoked Article 50 of the Treaty on European Union (TEU) which began the member state's withdrawal, commonly known as Brexit, from the European Union (EU). In compliance with the TEU, the UK gave formal notice to the European Council of its intention to withdraw from the EU to allow withdrawal negotiations to begin.

That is at this link. Starting the clock strengthens the negotiating position of the E.U. and so the E.U. had the incentive to require that the clock to be started before participating in real negotiation. The U.K. understood this.

The other answers rightfully analyse the political aspect of the question, but let's not forget that more fundamentally it would have been a denial of democracy to postpone the process indefinitely without a really good reason: the role of the government is to implement the will of the people, and the people happened to choose to leave the EU.

It's also worth remembering that at the beginning of the process, the UK government clearly overestimated their hand in the negotiation. They assumed that the EU would be accommodating for the sake of preserving its economy, and this was a massive miscalculation: the EU cannot legally offer an advantageous deal to a non-member state without offering the same to any other country. There was also a belief that the EU countries would not stand as a unified front, illustrated by some initial attempts by the UK to discuss with individual members instead of the mandated EU negotiation team. The UK government was also overconfident with the Irish border, an issue initially dismissed as easily solvable with technology.

Overall, the UK government appeared to be very optimistic about these negotiations, so they didn't see any reason to wait.

There was no plan to begin with.

When David Cameron initiated the Brexit referendum, it was done mostly/solely for domestic reasons, i.e. to calm the Anti-EU wing of his party.

After the Brexit referendum yielded the Leave result, Cameron resigned and Theresa May took over. Theresa May is well remembered for her "Brexit means Brexit" statement she made shortly after assuming the position of Prime Minister.

"Brexit means Brexit", in fact, tells you all there is to know about any plans or positions. Imagine your ideal Brexit, voila.

That's the reason why the Brexit negotiations are seen as confuse and chaotic from a EU perspective, as the British government, and the Tory party in particular, needed to arrive at a common view on Brexit. Something, that has not fully happened ever since.

Any answer to this is going to be mostly guesswork, I think, even if people offer very insightful analyses; I suspect that nobody really knows, except possibly the PM. It seems clear that there was a lot of myopic ideology involved in both pushing for the referendum, the subsequent leave campaigns and the unrealistic expectations to UK's negotiating position. One symptom of this, IMO, is the shrill statements about "The Will Of The People" - when the reality is that the result was what most would consider a very close call: 51.9 for leaving vs 48.1 for remaining. Hardly a decisive result.

So, to try to answer your question: I think the main reason was that the Conservative Party is deeply split over this and other issues, and the PM, facing an all-out civil war in her own party at a very critical moment for the UK, allowed the brexiteers to scare her into triggering article 50. I should qualify this a bit: I don't think Mrs May is scared of losing her position, but rather of seeing her party break apart and losing the government to Labour.

As with most things political it's about narrative. If you swallow the narrative that the UK elite genuinely wanted the UK to leave a functioning EU then it all seems a little odd. But what if the intention was to break the EU altogether? What if the intention was to collapse the EU project by starting a domino effect. Then events seem to make more sense? The UK is attempting to cause maximum disruption.. stir up discontent in other countries.. and cripple the EU. What if the UK elite had reason to believe that was all possible in a couple of short years? It turns out... It's not possible.. and now the UK risks loosing it's seat at a still functioning EU table.. with all that entails. The problem is that the EU is rallying.. and may (if my hypothetical narrative is correct) take a very dim view of the UKs actions.. Brexit may now be innevitable.. because the EU wants rid of perfidious Albion.. not because Rees-Mogg wants it.