Structure of public forum
Public forum debate is fundamentally a competition to convince a judge to vote for you. Everything in debate functions as a way to make the judge more likely to vote for you, and less likely to vote for your opponents.
Read the public forum Wikipedia page! I wrote most of it after a prolonged edit war, and it should help you out.
PF debate centers around a bimonthly or monthly resolution. For the months of September and October, as well as the months of November and December, one resolution is used for two months. Following this, the resolution cycles monthly, from January to April. (Later national tournaments, notably NSDA and NCFL Nationals, will have their own resolutions.)
Each resolution either asks whether some action would be justified (the US should abolish the capital gains tax), asks debaters to weigh the costs and benefits of some current action or process (on balance, the benefits of the internet of things outweigh the harms of decreased personal privacy), occasionally asks the debaters to evaluate a value or historical judgement (sanctions are reducing the threat Russia poses to Western interests), or occasionally asks whether one thing should be prioritized over the other (the US should prioritize debt reduction over economic growth).
A list of resolutions can be found here under "past topics."
At the end of every round, the judge will decide whether they want to cast a vote for the affirmative or negative team (each side of the resolution). Only one team may win, there are no draws. They will also award speaker points to each debater. (Most tournaments allow the judge to give different points for each individual, but WACFL only allows the same points for each team.) These points go up to 30, with the range varying by tournament. For most judges, points around 27-28 are average, but this varies by judge.
At the end of the tournament, teams are ranked based on their record (how many wins, how many losses) and their cumulative speaker points.
In some tournaments, sides and speaking order is not predetermined. In these instances, before the round begins you will flip a coin with your opponents. The winner of the coin toss gets to pick either their speaking order (if they want to go 1st or 2nd) or their side (if they want to affirm or negate). In general, it's advantageous to pick speaking second, as there are several advantages (as your opponent preps for speeches, you can prep as well, and you get the final word in 2nd final focus).
Public forum debate rounds involve alternating speeches and crossfires. Through these, you will convince the judge to vote for you.
There is a 1st team and a 2nd team, which depending on the tournament, is either predetermined or based on a coin toss. Every speech has the 1st team give theirs, and then the 2nd team right afterwards. (Be careful not to mix this up with 1st and 2nd speakers, who are on the same team.)
Constructive speeches: 4 minutes each
In the constructive, the 1st speaker will read their pre-written case. This case lays out the reasons the judge should affirm or negate the resolution. The 1st speakers are not expected to interact with the arguments of the other team in their own constructive, but instead only give the reasons to vote for them.
1st crossfire: 3 minutes
In this first crossfire, the two 1st speakers will ask each other questions to poke holes and gain concessions from their opponent. This is an important crossfire to set up the 2nd speaker's rebuttal by gaining key concessions, and get clarifications about the other team's case. As in any crossfire, making the opponent look dumb and yourself look smart is a crucial goal, informing the judge's perception of team dominance in the round.
Rebuttal speeches: 4 minutes each
In the rebuttal, the 2nd speaker will respond to the opposing case. They will explain why the arguments laid out in the opposing constructive are incorrect or flawed, and why their side is preferred in the round. Be sure your rebuttal speech isn't 100% answers, as arguments about the importance of your own case need to be emphasized. Make sure your responses to your opponent's case are grounded in logic, with clear reasoning backed up by evidence, not solely composed of it.
2nd crossfire: 3 minutes
In this second crossfire, the two 2nd speakers will ask each other questions largely focused on the previous rebuttal speeches. Debaters will aim to undermine responses made to their own case, as well as strengthen the responses they made to their opponent's. This crossfire should be coordinated with the 1st speaker to set up their summary speech, which is considered the hardest in the round.
Summary speeches: 2-3 minutes
In the summary, the 1st speaker will sum up the main reasons to vote for them, based on how the previous speeches have gone. In this speech, the debaters will collapse on their strongest arguments, focusing on only a few rather than all the arguments made in the debate so far. Teams need to focus the judge on the most important parts of the round, and ensure the judge understands the reasons they're voting for them.
A note: in 2019-20, the NSDA is piloting rules to change the summary speech to 3 minutes, which is adopted differently depending on your region. Make sure you know how long this speech will be before you attend tournaments, so you can practice correctly!
Grand crossfire: 3 minutes
In this final crossfire, all four debaters will ask each other questions. Since the only remaining speeches are the two final focuses, new information and arguments are usually not brought up. Grand crossfire often ends up devolving into a shouting match, so it's a great opportunity for you to present your arguments calmly, and make sure the judge doesn't get a bad impression of your team at the end.
Final focus speeches: 2 minutes
In the final focus, the 2nd speaker will give the judge the coherent reasons to vote for them. This speech is similar to the summary, and importantly, no new arguments can be introduced that weren't in the summary speech. After the final focus, the judge should have a clear picture of what voting for your side is, and in their reason for decision, your final focus should be mirrored. Emphasize the most important elements of your final focus to make them stick.