How should I be thinking about items when I play this game? Are the recommended paths any good? How do I know what to build?
Some are some aren't.
Be flexible with items.
In general;
Start with some regen items, some branches (3 is ideal, to build into a magic wand) or the 150g +3 stat items. Ring of protection is another option for early basilus ring.
Supports would like mekanism(mek), bracers, force staff, pipe (pipe of insight), wards, tranquil boots/arcane boots.
Offensive supports would like force staff, eul's, scythe of vyse, phase/powertreads/arcane boots.
Mid/gankers almost always want a bottle. You can store runes into empty bottles, once the rune is used you get a full bottle again.
Pusher/Gankers want drums of endurance, urn.
Make sure your team only has one urn or vlads or mek or pipe. Auras do not stack.
Carries want either power tread or phase boots, manta is popular for most agility heroes, desolator is good as well, helm of dominator if you want lifesteal orb. Strength heroes usually aim for the items like Assault Cuirass or Desolator.
Black King Bar is great on a lot of characters if you're facing a magic heavy team. Some spells do go through BKB though, usually single target ults.
Late game on carries you're looking at heart of tarasque, butterfly, monkey king bar, daedelus and satanic.
Skadi is rarely used, same with mask of madness. Mjolnir can be viable but not always. Necronomicon is used on heroes who either channel a lot or have skills that boost controlled units (Lycan/Beast Master).
My best advice is to read hero guides for specific items that are good for them. Some games I rush BKB other games I'll just build some heavier damaging items. Sometimes items make or break heroes. Faceless Void for example, does much better with Mask of Madness due to chronosphere then any other hero in the game.