Biggest Issues:
The adventure was written before 5e rules were really finalized, so you'll see some weird issues with rules being mentioned that don't exist, or encounters that aren't balanced.
Example: There's a random encounter early on, when the players are probably going to be level 4 or 5, that pits them against something like 4 assassins from the MM. Problem is, those assassins turned out to be CR 9 each.
Example: At the end of the second module, the players will encounter a Roper, which is way out of their league. The book (and later author clarification) talks about this encounter being a check for players to know that not all encounters can be won in combat. Problem with this is that the very first adventure that happened right before it opens up with "you come across a town under attack by an army of cultists and a blue dragon..." and expects you to dive right in. If anything, the players will have been conditioned to ignore potential warning flags for the sake of story based on previous events, which makes the roper encounter kind of bad. Secondly, there really aren't any other similar encounters to it down the road, where the players are encouraged to think carefully about what they can attack. They'll end up in one situation where they pretty much fight a vampire and white dragon back to back, and possibly a few giants unless they do some suspiciously metagaming roleplaying.
Recommendations:
Try and introduce some of the NPC's involved with the council of waterdeep in the second book, during the earlier adventures in the first book. Nothing crazy, but maybe have someone like Remalia meeting with Ontharr when they first meet him, and brief introductions can be made before she excuses herself. Or perhaps convert one of the random encounters that happen when traveling with the caravan to include a chance meeting with Delaan Winterhound (the one involving a stag is a decent choice, and could even be tied into the exiled elf prince story in book 2). It will make it a little smoother with the second module starts and the players are randomly thrust into a series of political meetings with important people that probably mean little to them.
Skip the random encounters during the trip from Baldur's gate along the trade way. Instead, pick a few that look cool to you, and maybe pull up an unrelated adventure that can take place during a few nights rest (wagon wheel breaks or something), and give the section some actual focus. See, the writers basically dump a bunch of NPC ideas on you, and say "these are the people that the characters can interact with" except they failed to provide encounters with enough context to support RP. So what happens in you struggle to paraphrase long stretches of boring travel time with a random encounter, and then you remember that the players probably need to RP a little and so you have some weird in-character conversations over a campfire with NPC that serve no real purpose. This is also a great place to introduce an NPC who travels with the players for a bit, and is really a metallic dragon in humanoid form. Perhaps he/she goes on one o the side quests with them (again, while a wagon wheel breaks or something), and gives the impression of being more than he/she seems; like royalty or something. The person is obviously a righteous character, and make it plain that he/she seems to be judging the PC's actions. The point is to set this NPC up as a tie-in to one of several random-but-really-shouldn't-be-random adventures that happens late in the second book, involving the players trying to gain the support of metallic dragons for the upcoming battle.
Take your time planning for the Castle Nayatar (or whatever the name is) adventure, because it has a ton of really fun potential built in. Yeah, they can straight up raid the place, but they can also incite a riot or pit rival factions against each other and use the chaos as a cover for infiltration. There's also a duo of black dragons in the area that the section mentions and immediately forgets about. Give them a purpose and maybe have an RP encounter with him (they pretend to be a single black dragon, but it's really two) to bargain for something, or maybe recover some item looted by cultist or who knows; just make sure you try and introduce the dragon(s) if for no other reason than to give them a presence in the AP. You can make it obvious that attacking them is stupid (probably is for their level), but it could also be fun to have them somehow successfully slay the dragon only to later find out it "didn't die" (due to it really being a duo) and now they have a hated enemy later on. Something like that. Remember, at this point in the AP, dragons aren't just mind controlled by the cult to do its bidding; they are vain and proud and selfish and probably don't think what the cult is doing to be very realistic, so it's not like they are throwing their weight behind it or anything.
Castle Nayatar is also the best place to really hammer home the point that some Red Wizards are helping the cult out, but the arrangement is obviously strange. I personally found Azbara Joss makes for an interesting 'double agent' who's sent to keep tabs on Rath Modar, get close to him and find out his plans, and -- if possible -- kill him if those plans work against the Lich King of Thay (can't remember his name spelling). To this end, Azbara could be a good future contact or interrogation target for information that becomes much more important in the next book, including the split Thayan faction things. Without this kind of change, the Thayan aspect of the story is largely peripheral until a single adventure late in the second book that kind of happens out of the blue and without much context.
Skip the vampire encounter on the cloud giant castle entirely. It's superfluous to that part of the story, and really doesn't make much sense at all to have a vampire essentially being on guard duty like that. I'd also pull the white dragon out because not only will you be fighting another white dragon a few adventures later, but it almost feels like the writers just included it because they felt like they needed a dragon fight *somewhere* in the first book rather than it enhancing the story. Instead, play up the danger posed by the cultist on the castle, and consider using Rezmir and Rath Modar together for a boss fight, supported with several waves of ogres and cultists to dispatch every few rounds. There's still the opportunity to RP with the giants, or fight them.
The tomb of Diderius is *very* interesting and although a bit random, can go a long way to providing a way for you to hint at (or even explain) how devils are involved in this whole thing (basically the major players in the nine-hells wants tiamat out and are willing to help in the ways they can). It's not built to explain the nine-hells thing, but it does include an out of place encounter with some bearded devils that came with Varen, and might seem a little odd to the characters. With some fun RP, you can use this as a chance to fill in whatever blanks you feel the players are having with the story.
I found the Sear of Moving Ice adventure to be pretty fun as-written. The only thing I didn't like was the Arcane Brotherhood being introduced as yet another faction in an AP that's already overflowing with factions. Make her a renegade harper or just part of some Waterdeep Research Academy or something, rather than hinting at more important people that ultimately will have little impact after that adventure.
Lastly, make good use of the "Cult strikes back" things, where the cult starts sending hit squads against the players. I'd probably not go the route of using dragons as part of the encounter, but maybe that would be fun in your campaign. Also remember that the cult isn't stupid, and they should probably know that hitting the PC's head-on isn't the best action (at least after the first failed attempt). The book lays it out in a way that it means they just throw more assassins and dragons at you, but I don't see why they could instead choose to start targeting people that the PC's care about; people that would be easier to kill.
