Although Miniman's answer seems the most straightforward, there seems to be one thing most answers on this page are not taking into account, which is the RAI in this Sage Advice Compendium entry:
Can you also cast a reaction spell on your turn? You sure can! Here’s
a common way for it to happen: Cornelius the wizard is casting
fireball on his turn, and his foe casts counterspell on him. Cornelius
has counterspell prepared, so he uses his reaction to cast it and
break his foe’s counterspell before it can stop fireball.
The casting time for Counterspell is "1 reaction, which you take when you see
a creature within 60 feet of you casting a spell," and it is a "3rd level abjuration."
This example has a wizard casting a spell as an action, and as a reaction, and neither are cantrips with a casting time of 1 action.
This contradicts and calls into question the the text in the Player's Handbook regarding how many spells can be cast in a turn, or round.
It also contradicts Miniman's statement "So if you cast a bonus action spell and a regular action cantrip, then trigger an AoO in the same turn, you would not be able to use your reaction to cast Shield."
Perhaps the text should read:
You can't cast another spell during the same turn,
except for a cantrip with a casting time of 1 action, and/or a spell with a casting
time of 1 reaction.
Additionally, I agree that the use of "a cantrip" is ambiguous in that it could be read allowing multiple cantrips, and that reading isn't game-breaking. With that reading you have bonus + action + reaction + action surge + action surge = 5. I don't think it would be game-breaking to take it one-step further (which would also simplify things) by saying "You can cast one spell with a casting time of bonus action, one spell with a casting time of 1 action, one spell with a casting time of 1 reaction, and any additional spells which are cantrips with a casting time of 1 action." I say "simplify," because I have seen in play many casters lose track of this anyway, and cast an extra spell they shouldn't have with no one noticing.
I am not sure if any abilities can give you more reactions, but if there were such things, given the above ambiguities, it seems you could cast more spells that way also.