Are you familiar with the MCU but not with AoS? Here's what you need to know:

2013 Fall
» Coulson lives! Agent Coulson is given a hand-selected team and a fancy plane (the Bus) to investigate weird stuff that pops onto SHIELD's radar. Daisy Johnson (then known as Skye), a hacktivist working with the Rising Tide to share information with the public, is picked up by the team for being a general pain in the ass and Coulson asks her to join them as a consultant.
» Project Centipede is an Extremis-containing serum being tested as a way to create super soldiers... There are explosive results.
» The team helps with cleanup post-Thor: The Dark World. They have run-ins with a handful of Asgardians (including Lady Sif) and a Berserker weapon. (Yes, it turns people into Berserkers.)

2014 Spring
» Coulson learns that he was brought back to life days after his death in Avengers by Project TAHITI, using a drug derived from Kree genetic material.
» HYDRA is revealed within SHIELD in Captain America: The Winter Soldier. Coulson and his team go on the run, discover one of their own (Ward) is a double agent, and help take down another double agent who is going crazy from his own use of the Kree-based drug. Coulson is appointed Director of SHIELD by Nick Fury and given a base from which to rebuild SHIELD in secret.

2014 Fall
» SHIELD continues working to track down and take out HYDRA cells while investigating the strange symbols Coulson has begun obsessively carving as a reaction to the Kree-based drug that brought him back to life. They discover this is connected to HYDRA's own interests, as well as to Daisy's search for her parents.
» SHIELD and HYDRA converge at the site of an ancient alien city where Daisy undergoes a transformation that unlocks her Inhuman heritage and gives her the power to manipulate vibrations.

2015 Spring
» Daisy is taken to an Inhuman refuge called Afterlife, where she is reunited with her parents. Her mother, who was tortured by HYDRA (an incident that tore their family apart), blames SHIELD and uses her position as leader of Inhumans to start a war with SHIELD and humanity at large. This ultimately results in her mother's death and Terrigen being released into the ecosystem, which begins triggering transformations in other potential Inhumans around the world.
» Agent Jemma Simmons vanishes through an alien monolith that was formerly in the possession of HYDRA.

2015 Fall
» SHIELD tracks down recently-transformed Inhumans to help them learn to control their powers and start a new life; some of these Inhumans join a special SHIELD team led by Daisy. The government forms the Advanced Threat Containment Unit (ATCU), which also tracks down Inhumans, and The Watchdogs terrorist group forms with a mission to kill all Inhumans.
» Simmons is brought back from the alien planet she was transported to by the alien monolith, where an Inhuman known as Hive destroyed all life. Hive was worshipped for thousands of years by a religious cult that became HYDRA, with the intention of one day bringing him back to Earth — which occurs when Hive takes over Ward's deceased body.

2016 Spring
» Hive attempts to recreate the Kree experiments that created Inhumans thousands of years ago. He uses his parasitic abilities to sway Daisy, brainwashing her into becoming one of his army, and plans to transform the human population of Earth into primitive Inhumans. SHIELD stops him at the cost of Daisy's boyfriend's (Lincoln) life.
» The Sokovia Accords are ratified in Captain America: Civil War.

2016 Fall
» Daisy works to take down the Watchdogs, who continue to terrorize and kill Inhumans. She has left SHIELD and works as the vigilante Quake, who frequently appears in the news as she travels across the country.
» Ghost Rider! Robbie Reyes teams up with Daisy and SHIELD when they discover his uncle is using the Darkhold.
» SHIELD becomes a publicly-known agency again with a new "Inhuman" director in charge. (He's faking it.) Daisy rejoins SHIELD after the director claims her vigilante exploits were undercover work. She speaks to a congressional committee about Inhumans.

2017 Spring
» AIDA, a Life Model Decoy (super advanced robot), reads the Darkhold and becomes corrupted by its power. She creates other LMDs and begins replacing agents within SHIELD, trapping their minds within the Framework, a simulated world.
» Daisy and Simmons enter the Framework, where the world has been reshaped into a What If nightmare where HYDRA won and Inhumans are openly hunted by the government. They rescue the team and take down AIDA, but not before she causes SHIELD to end up on the government's shit list again.

2017-2018
» Team SHIELD is transported to a post-apocalyptic future where the Earth has been torn apart and what is left of humanity has been enslaved by the Kree. Daisy is blamed for this and dubbed the Destroyer of Worlds.
» Upon returning to the past, the team works to stop this future from coming to pass. General Talbot, a former (slightly antagonistic) ally, absorbs gravitonium through a process developed by a last vestige of HYDRA and becomes the villain Graviton. Daisy takes an upgraded version of the Centipede serum and defeats Graviton.
» (The fight with Graviton takes place around the same time as the fight with Thanos. There are actually warnings given to various people, Daisy included, that Thanos is coming, but it's not clear on the exact timing of Thanos's arrival and the Snap.)
» Following Coulson's final death, Daisy and Simmons leave Earth to search for Fitz in space.

2019 Summer
» One year later, SHIELD is back! An alien with Coulson's face arrives with his freaky biker gang to stop parasitic alien birds that herald the arrival of the being Izel and the end of the world.
» Daisy returns to Earth after they find Fitz, there are emotions and weirdness with not!Coulson, and the world almost ends before they defeat Izel.
» A species of alien robots called Chronicoms decide to conquer the Earth and claim it as their new homeworld after Izel destroyed theirs. Since SHIELD is the main resistance to their plans, they go back in time to take out SHIELD in the past so they can overtake Earth in the future. An advanced LMD of Coulson is created to help with this mission.
» Team SHIELD travels through the past, hitting 1931, 1955, 1973, 1976, and 1983. They run into a young head of HYDRA, rescue Daniel Sousa from assassination, and avert Project Insight from launching 40 years early, among other exciting adventures.
» Ultimately, they fuck up the timeline a lot and most of SHIELD is wiped out by the Chronicoms. The team travels back to their original timeline via the Quantum Realm and defeats the Chronicoms by Daisy blowing up their fleet from the inside — she dies, but she gets better.

2020 Summer
» One year later, SHIELD is still going strong. They have a helicarrier and a new Academy named after Coulson. Daisy is leading a team on a spaceship as they explore the universe.

The Multiverse Consideration: Since Marvel refuses to clarify whether AoS is actually within the film timeline of the MCU and some people feel very strongly that it is not, I'm totally fine with treating AoS as existing in an alternate timeline and crossing over to the film universe. This is within AoS canon as of season 7 so it's not a problem to integrate into the story!

(But also, please consider this.)