I know I'm about to face a bunch of enemies, I have to draw them out first. If I know I'm about to be in a huge battle, can my weapon be drawn before I go in there or do I have to wait until the encounter actually starts? It seems to me that walking around in a dungeon with my weapons put away is kind of silly, especially when I know there will be multiple encounters in the near future.
Your weapon can be drawn before you go in.
There's nothing to say that you can't walk around holding your weapon, though provision of light and carrying of gear are common reasons to have sheathed a weapon. (Social nicety is another.)
When can you draw it? Out of combat you likely just declare your draw and GM says "okay." In combat, on your turn, you can draw you weapon as an interaction with an object.
PHB p.190, "Other Activity on Your Turn"
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
(Note that this interaction is a "free" one in that it does not consume your action.)
On the same page is a table with about 20 examples of "Interacting with Objects Around You." Drawing a weapon is the first one mentioned, picking up a weapon that was dropped is another.
Specific restrictions might beat this general rule, but this is the general rule.
Abso-positively. Go ahead, weapon-walker.
The answer to this has more to do with why you would carry your weapon around rather than if you can. There are various reasons to do so, including walking around in a combat-heavy dungeon or expecting an ambush, but there are actually more reasons not to do so. The other two common components of dungeons, traps and puzzles, both are better suited to a weapon sheathed; having to let go of a weapon to grab the edge of a pit trap is a very unfortunate way to derail the group, and many puzzles require the use of hands. Social situations get out of hand when someone has a weapon drawn (though social encounters and combat rarely grow from one another, unless you really ticked off someone...). Overall, the answer is yes, you can carry a weapon in hand, but it might not always be best. That is why you might not carry it around all the time in a dungeon.