When running a module don't be afraid to tell the players that they should expect things to happen within the confines of the module. That's not railroading, that is just playing a cohesive game. I find that many new DMs are so afraid of railroading that they overcompensate into the other extreme. Then that adventure becomes an incoherent mess which also isn't fun.

As for Dragon Heist which I have run recently I got a few tips: You can run it as is which imo is absolutely fine and I did that. There is also the "Alexandrian Remix" which expands the module fairly short module to a huge campaign adventure. I personally don't think you should do that as a first timer but it's still useful to know it exists.

From here on very major spoilers. Players of the module don't read further.

Before session 1 what I did was give all players the Waterdeep Enchiridion to use as player knowledge if they want to. 2/4 players red most of it. You as the GM should read chapters 1 to 3 and the Enchiridion fully, plus the major NPCs. Chapter 4 only makes sense once you figured out which villians to run. I selected 2 villians to have appear. Imo more is too much to keep track of for the players in this context and they already felt overwhelmed at times. I used the Casselanters as main villians and the Xanathar guild as further Antagonists. Then I had the Zhentarim and Jarlaxle only slightly involved in the background and as a faction in chapter 2.

Chapter 1 is fairly linear. Bar fight, troll comes up (I changed the troll to a devil to hint at infernal activities by the casselanters). Quest by volo to investigate. Absolutely use Old Xoblobs shop, it is very fun. Investigations in the Skewered, encounter in the streets. This is where I'd level up the party to level 2. The encounter in the warehouse with the Kenku is pretty deadly otherwise. And the Xanathar hideout even more so. This is the biggest balancing flaw in the whole adventure imo.

Capter 2 starts once they got the tavern. I leveled them up to 3 at this point for 2 reasons. First so I can run more faction missions in this chapter so the fireball chapter doesnt get diluted as much. Secondly because it felt like the correct timing. This chapter is the open sandbox part. Stay here as long as it is fun. Let them get to know the neighbors, decide weather you want the competitor guy interfere with the tavern, how far you want to take the management part, or if you let them higher a manager etc. The ghost can be fun too. There is a fantastic DMs guild resource with fully fledged out faction quests for chapter 2 which I used. This is where my players did a ton of cool shit. They made friends with the Zhents and Black Viper, Once this stops being fun throw the fireball.

Chapter 3 can be mostly run as is imo. You might need to hint less subtly at how to find the construct. I didnt like the part with the detection device so I had the group follow some other clues. Also I had Floxin in liege with the Casselanters because I removed Manshoon.

Chapter 4 against all advice I ran exactly as is with the chase rules from the DMG, it wasn't even 2 sessions of 3h each long. But that was the most fun part of the adventure for me. The players got the stone but had no idea who the villian was yet. So they ended up handing the Stone to the Casselanters asking for 10% of the treasure which they did get (50k gold is nothing to sneeze at).

Chapter 5 the players Casselanters were invited to the Founders Day Festivities where the sacrifice was to take place. I took some inspiration from the book but mostly made up the exact events as it fit to what happened so far.

Dragon Heist is the most fun I had running a 5th edition module so far. So I hope you enjoy it as well and I was able to help a bit with your preparations.