Are there any rules that specify whether or not switching weapons in combat takes up an action?
Yes, there are.
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.
If you want to interact with a second object, you need to use your action. Some magic items and other special objects always require an action to use, as stated in their descriptions.
[PHB, Chapter 9: Combat, The Order of Combat, Your Turn, Other Activity on Your Turn]
A shield takes 1 Action to don and 1 Action to doff. [PHB, Chapter 5: Equipment, Armor and Shields, Getting Into and Out of Armor]
Many GMs allow you to carelessly drop an object at any time (it makes sense not to require any Action to just open your hand and let go).
So, in your question, it would take a round to doff the shield before you could even start to pick up a bow and use it.
You can stow or draw a weapon as a free interaction while moving or making an attack. Dropping a weapon is a valid option. (Had a fun moment in my last session where my barbarian had stowed a handaxe on one turn and dropped the other (mic drop style) before pulling out his greataxe while tilting his head to the side to really let those goblins know that it was on.)
So, in your question, it would take a round to doff the shield before you could even start to pick up a bow and use it.
The other issue here is that there's no sensible way to carry a bow (or most of the weapons with the two-handed property for that matter) on your body - especially if you've already got a backpack full of adventuring gear on your back. If it's not a one-handed weapon you're most likely going to be carrying it by hand (and even then, one-handed pole weapons like spears and quarterstaves had to be carried by hand too.)
Are there any rules that specify whether or not switching weapons in combat takes up an action? For example:
If I'm wielding a shortsword and shield one turn, can I make a bow attack next turn without using an action to change equipment?
Could I switch the shield with another shortsword and still make two attacks with the shortswords?
Yes, there are.
[PHB, Chapter 9: Combat, The Order of Combat, Your Turn, Other Activity on Your Turn]
A shield takes 1 Action to don and 1 Action to doff. [PHB, Chapter 5: Equipment, Armor and Shields, Getting Into and Out of Armor]
Many GMs allow you to carelessly drop an object at any time (it makes sense not to require any Action to just open your hand and let go).
So, in your question, it would take a round to doff the shield before you could even start to pick up a bow and use it.
You can stow or draw a weapon as a free interaction while moving or making an attack. Dropping a weapon is a valid option. (Had a fun moment in my last session where my barbarian had stowed a handaxe on one turn and dropped the other (mic drop style) before pulling out his greataxe while tilting his head to the side to really let those goblins know that it was on.)
The other issue here is that there's no sensible way to carry a bow (or most of the weapons with the two-handed property for that matter) on your body - especially if you've already got a backpack full of adventuring gear on your back. If it's not a one-handed weapon you're most likely going to be carrying it by hand (and even then, one-handed pole weapons like spears and quarterstaves had to be carried by hand too.)
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