The maximum spells you can cast in a round is 3. Generally it's going to actually be only 2 on your turn, since reactions are usually outside of your turn.
On page 159 of PHB:
You can only take one bonus action on your turn, so you must choose which bonus action to use when you have more than one available.
Quicken Spell, Action Surge, or a spell that has a normal casting time of 1 Bonus Action all require your bonus action for the turn, so you can only choose one of them each turn.
On your turn you can:
Move up to your max movement, reduced by various factors and broken up between actions
Perform 1 Action
Perform 1 Bonus Action if you have an ability that allows you to use it
Perform as many Free Actions as your GM allows.
You also have 1 Reaction per round, which is generally outside of your turn, but doesnt have to be.
The book makes this very clear with no room for misinterpretation.
