I was reading through the provided Black Fang Adventure and the room with the big spider in it, I was confused about the big webs in the corner. Should I make note of them on the map like I would difficult terrain or should I only bring up the STR check only once they attempt to leave one of those squares?
Also, several of the rooms have a treasure chest icon on them. In one room it is a goblin body, in another an actual chest. Should these details be written on the map as the PCs enter the room?
Unrelated to the maps, but related to the treasure, what about the items is common knowledge? The items with magical properties, I understand I would want to encourage players to use Detect Magic, but how should I describe the magical items as they discover them?
I have not looked at the beginners box, but if the players can see it put it on the map. Players will feel cheated if they get caught in an invisible web, or miss a large wooden chest because it wasn't shown to them. Whether you put it on the map or not TELL your players that it is there, not everyone is a great artist, and a web or cheat may not be self evident in the art
In general you should only make people do perception checks if something is not obviously visible.
That giant troll that's standing in their way 20 ft away? Yes, they probably see that, so just tell them without a check.
The chest thats hidden in a secret compartment behind the bookshelf? Not so much.
If its dark in the room, or they're busy fighting stuff when they enter the room, it might be appropriate to call for a perception too, as they're not actually paying full attention to their surroundings.
Creatures actively trying to hide (stealth check) also need to be spotted first by an oposed perception. So the same troll trying to hide behind a tree might not be that obvious anymore.
Yep basically what Quatar and Ubercroz said.
For the detect magic thing just describe it in a way that maybe makes you think ohhhh magic. It's not so important later on in the game to describe things like this but that first vial will kind of give the idea that anything could be magical.
Have the potion glowing or it feels warm to touch etc.
To add to what others have said, it's all in how you describe something to your players. Maybe the weapon or armor has finely scribed writing and is free of dust, even though it has apparently been sitting for a long time, objects could glow, or explain that the crafting appears to be of a better quality than one would normally expect - and it doesn't need to be much, just something to set it apart from the other crap lying around.
As far as the map (especially using a pre-printed map like the flip-mat), if you want to keep the parts of the map hidden that the players have yet to discover, you can cover those portions with post-it notes. As the players explore, remove the post-its out to the extent of their vision (or the confines of the room).