Does picking up a weapon or drawing one take an action?
No, drawing a weapon does not take an action. Nor does picking up a dropped weapon. Both answers are found in the "Interacting with Objects Around You" sidebar on page 190 in the Player's Handbook (PHB):
Here are a few examples of the sorts of thing you can do in tandem with your movement and action:
- draw or sheathe a sword
- pick up a dropped axe
Note that you are allowed to interact with only one object per turn for free (p. 190, Other Activities on Your Turn, PHB). This means that you can draw or sheathe a weapon for free as part of another action (such as a move or attack), but to do both in the same turn you'll need to take the Use an Object action or have the Dual Wielder Feat (p. 165, Feats, PHB).
Note that the Dual Wielder feat only lets you draw two weapons or sheathe two weapons when you'd otherwise draw or sheathe just one; technically, you can't both draw a weapon and sheathe a weapon (whether it's the same weapon or a different one) with a single object interaction, even with the feat.