The highest level element of Brexit occurred on 29th March 2017, when the Prime Minister, Theresa May, sent a letter to the President of the European Council, Donald Tusk, formally notifying him of the UK's intention to withdraw from the EU, in line with Article 50(2) of the Treaty on European Union, aka the Lisbon Treaty, and responding to the referendum vote the previous year.
Due to a UK Supreme Court decision in the government appeal of the High Court decision in R (Miller) v Secretary of State for Exiting the European Union, this had required parliamentary approval, which was obtained in the form of the European Union (Notification of Withdrawal) Act 2017.
Since the UK was intending to leave a large number of EU institutions, and since EU law would no longer apply to it, it was necessary to modify a large amount of legislation which would otherwise become meaningless or ambiguous, neither of which is a very happy state to be in. To this end, parliament passed the European Union (Withdrawal) Act 2018 which both stated that many laws were to be frozen in the state that EU law and regulations had left them, and gave Government ministers time and extent limited powers to rewrite legislation to make sense.
A bill like this was going to be necessary, regardless of whether a exit agreement (a "deal") was obtained, purely to recognise the constitutional changes Brexit will enforce. After a very long game of ping-pong between the two houses of the UK parliament, the bill received Royal Assent on 26 June 2018.
In the meantime, civil servants from both the UK and EU have been meeting, with oversight from elected politicians, in order to attempt to generate a full formal exit agreement. From the outside, these meetings have appeared acrimonious, with leaks from the EU side frequently suggesting that the UK government was unsure what it actually wanted. The status of the land border between Northern Ireland and the Republic of Ireland has been a particular sticking point. While a theoretical "backstop" was agreed which ought to ensure a "frictionless" border, the one thing both sides have both stated is that "nothing is agreed until everything is agreed".
At the same time, a raft of primary legislation has been going through parliament to give the government powers to deal with matters not dealt with in the big Withdrawl act. At times this has exposed divisions on Brexit in both the Conservative and Labour parties, with rebels from both sides common when legislation seemed to be favouring a hard or soft Brexit. In an attempt to limit this, on the 12th July 2018 the Government released the Chequers plan, formally the white paper on "The future relationship between the United Kingdom and the European Union". Which summarised the position she expected her cabinet to accept. In response David Davis, the Brexit Minister, Steve Baker, his junior and Boris Johnson, the foreign secretary resigned. The EU negotiating team have also rejected the plan as unworkable.
While it's possible parliament will vote on its approval of the white paper, this doesn't in of itself has much force. More importantly, as part of the horse trading to get the EU Withdrawal Bill passed, they promised parliament a vote on any exit agreement, before it was put before the EU ratification process (in part due to the agitation of the Conservative MP Dominic Grieve). However, language in a Lords amendment, which would have ensured parliament had the power to direct the government what to do in the even the bill was voted down(the so called "meaningful vote"), was rejected in the Commons. As such any vote on such a bill (with the proposed name The European Union (Withdrawal Agreement) Bill) is likely to be "take it or leave it".
In the constituent countries with working devolved parliaments (i.e. Scotland and Wales) there is similar legislation to the Withdrawal Act dealing with references to EU law in their own legislation. There is also an ongoing tussle over whether ministers have the power to modify legislation in areas which the devolution legislation has promised is theirs to control.
This answer is rather longer than I originally intended, but Brexit is a perfect example of "politics as soap opera" in which identifying key elements is difficult.
