GTNH relies heavily on the Better Questing Quest Book. You can play the pack (with small exceptions that won't matter to you now) completely without doing the Quests, but you would miss out on a lot of the content. The quests also guide you through the progression, explain how various machines and mechanics work and give you hints on cool things to explore at your stage in the game. So don't miss out on this opportunity! Oh, and you also get cool rewards.
You can open the Questbook either through the item you got at the start or, much easier, through a hotkey. You find this in the Controls menu under "Open Quests", also make sure there are no conflicts on the key you choose or it might not work.
You don't need to complete every quest to progress. Some quests are marked as main, those mark the progression path and you will need to complete those to unlock new tabs and generally progress in the book. You can recognise them by the pointy corners. Main quests are also connected with thick lines. You will find information on various topics and explanations on the game mechaincs there, so be sure to read them. The other quests are optional and provide you with ideas on what to do at your current stage or useful machines and items to craft. You don't need to complete those, but they are nice to have.
As you progress, you will unlock new tabs for the GregTech power tiers, tabs for specific mods and tabs with a shared topic for things which don't necessarily all belong to the same mod, but have something in common nonetheless.
When you open a quest you will see the required actions on the right. This can have several steps you can swap between using the arrows. You need to complete every step, so be sure not to skip anything. Some quests ask you to check in the ingredients before you are supposed to craft the final product. Most quests are Retrieval Tasks. To complete those, you just need to have the required items in your inventory for it to complete. You can also click the Detect button if it doesn't refresh automatically. Especially at the start, there are a lot of Crafting Tasks. Those require you to actually craft the item in question, it is not sufficient to simply have it in your inventory.
Most quests will reward you either items related to the theme of the quest or a lootbag, often you will have to choose. The lootbags will give you random things more or less relevant in your current tier or trash. You can disable the trash by enchanting the bag with Fortune 3, but this isn't really viable before you can dupe the books with BiblioCraft and even then it is very expensive and often not worth it. Additionally, you will get coins. Those can be used in the "Coins, Coins, Coins" tab of the quest book to buy various rewards. Those can save you some tedious crafting or provide you with new options. Some things to look out for are the oreberries, that can be planted on crop sticks for renewable basic metals, and ore samples for your ore finder wand for all the ores RNG doesn't want you to find.
Your team can join together in a questing party to share the quest progression. Note, that every quest can only be claimed once though. It is still more fun to play in a party, since a lot of effort is otherwise required to sync up your quests manually and do all the crafting quests once per member. See this as a balance thing, your team will have to cope with. If you persuade someone to join you in this trip to hell, cough, I mean adventure, after you already started, they can still join your party. They will have to complete all the retrieval tasks again though, until they are in sync with the rest of the team. Crafting tasks will automatically complete for them.
Sometimes you will find a quest that you can't figure out how to complete. If you need help, go to Discord and ask. If you are sure it is a bug, go to github and report, if it hasn't been reported yet. In the meantime, you can complete the quest for yourself using commands. You will need to be OP for this on a server and have cheats enabled in singleplayer. Alternatively, you can also Open to LAN for this and close again afterwards. You need the Quest ID for this. Hover over the quest in question and press shift. This will display you the number. Then you can run the command /bq_admin complete < Quest ID> <Your name>. Of course without the < and > and the quest ID without the leading #.
Remember to make backups from time to time, especially if you update. The Questbook can break from time to time, so you don't loose everything then. The relevant files are in your saves/<worldname> folder with file names starting with Quest. If you need to complete multiple quests because your backup is too old, you can run /bq_admin edit, open the book^, click Edit, choose a tab and click Designer. This menu offers you a Manual Complete towards the bottom on the right side. Using this you can easily complete quests by simply clicking them. If you are finished, just run the command again to exit the edit mode.
Modpack updates often expand the questbook or fix quest related issues. To profit from this, you need to run /bq_admin default load once after an update. This will refresh your quest database.