The maximum number of spells a character can cast in a round is 3. You can only take 1 reaction per round, although some monsters have an ability that allows them to ignore this restriction.
You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
This means that you can cast a bonus action spell and a regular action cantrip on your turn, and a reaction spell at some point in the round that is not your turn.
A round consists of every creature in the combat taking a turn. This means that you are free to use your reaction in the same round, as long as it's not on your turn.
It's very difficult to trigger a reaction on your own turn anyway, since most of the triggers for reactions are your character's reaction to something else. The only exception I can think of is moving such that you trigger an attack of opportunity, since being attacked is a trigger for a number of reactions.
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. But if you were attacked after your turn had ended, you would then be able to cast Shield.
Largely unnecessary caveat
There is one point which confuses the issue a bit - the Fighter's Action Surge ability.
You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
This can be interpreted as meaning either you can cast a single cantrip with a casting time of 1 action in the same turn, or, you can only cast cantrips with a casting time of 1 action in the same turn. If the former, the maximum is 3 as written above. If the latter, you can cast 4 as written in this answer.
Personally, while I'm not sure of the RAW answer here, if a player wanted to waste use their Action Surge on casting a cantrip I would definitely allow it.
Reference: PHB, p. 202.
Controlling a spell is not casting it
On the first turn, you are right, you can cast Spiritual Weapon (as a bonus action) and a Cantrip (as an action) but not any other spell.
On the subsequent turns you can cast any spell as an action and then use your bonus action to control an already existing spell. In fact, you can have both active and control one of them using your bonus action, while the other one is on standby and still cast another spell.
Note, that Bigby's Hand duration is based on Concentration. This means you will not be able to cast any other Concentration spells if you want to keep Bigby's Hand active at the same time. Spiritual Weapon does not suffer from it.
Best Answer
You were quoted the rules incorrectly.
First, Eldritch Blast has a casting time of 1 action (PHB p.237). You've only got one action on your turn, so you can only cast Eldritch Blast once. ("Off the shelf," that is. There are features, like the Sorceror's Quickened Spell, Action Surge, or even time stop that might change this.)
Boom. That's it. They can't cast a second eldritch blast as a bonus action, because the casting time of eldritch blast isn't "1 bonus action." (Compare with, say, healing word.)
But that's understandable.
It is possible for a caster to cast a spell as a bonus action and another as an action in one turn; the conditions are described exactly on PHB p.202:
D&D is a great game in many ways. It has never, in my experience, been presented in its published materials in a way that makes sense to someone who doesn't already know the game. One player misunderstood something, and no other players nor the GM knew the rule well enough to course-correct mid-session.
That's fine. You now know. Alert your group as soon as possible and be ready for a group conversation about whether this should now be a table-rule, or will this just have been one shining moment in 'Lock's career. "Hey, guys, remember that one day when I was in beast-mode? My patron must have been having me on!"