First things first - everything's fine
The players didn't save Sildar in Chapter 1, they went through the wolf cages and skipped it entirely so they don't even know he was in there. Is this going to cause any issues later on?
No, there are lots of ways to go from here when your party did not rescue this NPC at first. A lot of NPCs might point out the information Sildar would provide or could point the party to the fact that Sildar is still missing.
Also, the 2nd chapter ends with them inside Glasstaff's quarters, and the 3rd chapter seems to begin almost immediately in the Spider's Lair. Is there anything that needs to be done in between?
There is plenty to do between different chapters. You just need to carefully read the text of the third chapter and remember that the book is trying to be a guide to you DMing - not something you have to follow to the letter.
My party didn't rescue this NPC
When they arrive in Phandalin they should have the caravan with them. Someone will probably ask them about it. For example:
Why do you guys have this caravan? Most people with stuff like that go to Elmar Barthen first to do business with him. Who sent you? Remember that there were barrels with Lionshield Coster's emblem in the hideout. This NPC probably wants those barrels and crates back. Maybe you can place a few more there and send the party back?
Make the town come to life by letting an NPC use this chance to remind the party about why they were originally coming to the town.
Someone, probably Barthen, will then ask them about Gundren. And maybe he even knows about Sildar. You can use this to send the party back and retrieve Sildar if you want them to talk to him.
A lot of people know about Cragmaw Castle.
For example the Bugbears and Iarno Glasstaff Albrek should know something.
If your party captures one of them they can interrogate them.
If not then Sildar could maybe tell them something he overheared while he was captured. If your party didn't rescue Sildar, even after you dropped some hints, let him be rescued by someone else. The world is alive and not only revolving around your party.
This is a perfect chance to show them that they missed something by mentioning that Sildar is helping another small group of adventurers. Or maybe they just dropped him off and went away, but he told everyone to give them a place to sleep at his cost should they ever visit again.
What to do between Chapter 2 and 3?
If you read the text carefully you will see that the third chapter is presenting multiple dungeons - just smaller ones without a specific map. These are side quests you can use that are tied to Phandelver. For example
the party meeting with Agatha, checking out Old Owl Well or Wyvern Tor. The third chapter also mentions the ruins of Thundertree and of course Cragmaw Castle.
What to do between Chapter 3 and 4?
The fourth chapter begins with the Wave Echo Cave. But remember that the book is just a guide. How did they arrive there? Well, that is your decision and the book is making you comfortable with winging it.
Most of the ways that are intended can be read in the What's next? paragraph at then end of chapter 3 on page 41.
They could have a map and follow that one.
In chapter 3 they can get a map. Also there is a wilderness encounter table at the beginning of chapter 3 you can use and the beginning of the book contains a map of the region to show you how long they might travel. Perfect to train a bit for example.
They might have rescued someone that knows the location.
They might have rescued Gundren. He can show them around and lead the way without the fear of running into wild encounters, though you might still want to throw in a few.
They might have interrogated someone that knows something.
They might have asked King Grol in Cragmaw Castle or they managed to intimidate the Drow. The adventure says the Drow should not know - but you are the DM and if your players ignore other hints you can just invent one or two new ones for them to find.
They might have done some side quests
For example they might have gotten the information from Agatha, Reidtoh or Hamun Kost.
They might be blind - but some NPC will surely sell information for enough gold or a favor.
They might have neither Gundren, nor a map. Then maybe Sildar knows a few rumours that might point them in the right direction. Or some other NPC offers some information if they help with a previously ignored side quest. Or a merchant enters the town, selling a map.
They might be blind and killing everyone in sight.
They can always just start searching the wilderness and maybe they randomly come across it. You might want to place the cave somewhere else in that case so that they don't have to search forever.
It's allowed to fail in an RPG
Videogames for example have no way to progress if you don't do something. But in an RPG like DnD something happens all the time. The world is alive. If they never get to the cave - then so be it.
You can just place the Big Bad Guy somewhere else. Or let him come to the PCs. Maybe he wants something from them. Or heard that they were killing his minions and is seeking revenge. Or he just wants to control Phandalin now that Iarno is captured/killed/convinced of the good path/bribed/...