The most important things to keep in mind are the Cagewrights, the Shackleborn, and Vhalantru the beholder

To briefly (spoiler-ly) summarize of the plot of Shackled City: The Cagewrights are a group of 13 bad guys who have learned how to make an artifact to connect the city of Cauldron and the plane of Carceri, with the hopes of using it to free the demon lord that is imprisoned there. The adventure starts after the Cagewrights have built their artifact, but still need to collect people with a connection to Carceri (the Shackleborn) before they can start their plan.

The first 5 adventures introduce to the players that something strange and nefarious is happening in Cauldron, and involves them unwittingly harming and helping the Cagewrights' plans without learning any specifics. The sixth adventure, Test of the Smoking Eye, has them learn specifically about the Cagewrights, and afterwards the remaining six adventures consist of the players going on the offensive and trying to stop the Cagewrights' plans.

Thus, the first five adventures intentionally introduce stuff that the players aren't able to follow up on until later in the adventure. Additionally, many of these hints or pieces of information are easily missed, and without finding them the first adventures can feel a little directionless. Ensuring that the players find these hints and making certain characters pop up more frequently could help. Some of these plot points include:

Vhalantru the beholder

This is the villain that your linked review refers to. He appears at the end of Life's Bazaar (Adventure 1) and takes one the children the players are trying to rescue. The players realistically don't have a way to fight him, and the slaver Kazmojen who knows the identity of the beholder won't give it up even if charmed. The players don't encounter Vhalantru again as a beholder until Lords of Oblivion (Adventure 8), a very long time later. Vhalantru is secretly a lord in Cauldron, and the players run into his human form in Flood Season (Adventure 3).

Having the beholder appear or be mentioned more could bridge this connection, but note that Vhalantru is important for the progress of the plot (he eats the mayor off-screen for example) and is relatively well known in the city so you shouldn't have his identity revealed before Adventure 7-8 unless you're willing to rewrite parts of the plot.

The Cagewrights

The main villains of the adventure path go almost unmentioned in the first five adventures (which is probably my biggest issue with the adventure path as a whole). They all wear the same ring so the players can tell when they've killed a Cagewright. It's possible, though unlikely to hear the names or spot three of them in the first five adventures:

Lady Thrifirane Rhivaldi teleports in and out with Vhalantru at the end of Life's Bazaar, but is invisible and doesn't do or say anything. She doesn't come up again until Lords of Oblivion.

The previous leader (who is so forgettable I can't recall the character's name) of the Ebon Triad (the antagonists of Flood Season) has joined as a member of the Cagewrights. The current members of the Ebon Triad don't have a lot of reason to mention their old leader to the players, so it's unlikely they'll learn about it in the published adventure. The previous leader isn't encountered until Thirteen Cages (Adventure 10) or Strike on Shatterhorn (Adventure 11).

Embril Aloustini is the high priest of Wee Jas who is away from the city during these adventures, which the PCs may learn near the end of Flood Season. She appears for the first time in Strike on Shatterhorn.

The other ten Cagewrights aren't encountered until Thirteen Cages or Strike on Shatterhorn, where they likely die in the same encounter they're introduced in.

The Shackleborn

The Cagewrights need 13 Shackleborn to power their artifact and have most of them at the start of the adventure. Th players are involved with the Cagewrights collecting two Shackleborn: Terrem in Life's Bazaar and Zenith in Zenith Trajectory (Adventure 4). It's also possible for PCs to be Shackleborn if they take an optional trait provided in the hardcover edition. Shackleborn can be identified by an invisible birthmark (seen using ver invisibilidade for example). In 3.5e, there's a warlock invocation that grants unlimited ver invisibilidade, but without that it's unlikely Zenith's mark will be noticed and basically guaranteed that Terrem's will be missed. The players need to see the birthmark on both to understand any sort of connection, so you may want to make them easier to notice.

Small Stuff

The Last Laugh thieves guild is introduced in Life's Bazaar, and then doesn't come up again until Lords of Oblivion where they kidnap a character and their leader attends a recruitment meeting for the Cagewrights

Hooknose the red dragon and the Morkoth that lives in Crater Lake both attack the city in Foundation of Flame (Adventure 9). Hooknose's kid is encountered in Zenith Trajectory and can be mentioned in gossip, and the Morkoth has a tavern named after it plus it's mention in Flood Season.

The impact of NPC allies such as the Striders (met in Life's Bazaar and Flood Season), the Chisel (Zenith Trajectory and Demonskar Legacy (Adventure 5)) and the frenemies the Stormblades (Flood Season) is left up to the DM and players. The only NPC routinely pops up throughout is Jenya Urikas (Life's Bazaar).

The Magical Threats Agency is introduced in Zenith Trajectory and then never comes up again.

Celeste the innkeeper is introduced in Zenith Trajectory (where she is unknowingly helping the Cagewrights), and then is later trapped in a painting by Vhalantru which is found in Lords of Oblivion.

Nidrama the angel appears in Demonskar Legacy and then returns in Asylum (Adventure 12).