Round 1 - Cast Haste - no issue, here. Though you could squeeze in one more attack (possible Divine Smite-ed) after casting.
Round 2.1 - Cast Green Flame Blade, on a hit, use Divine Smite - Divine Smite only needs you to hit with a melee weapon attack, which GFB does. No problem.
Round 2.2 - Hasted attack with Divine Smite - Pretty standard.
Round 2.3 - Cast Thunderous Smite and hit - Nope! You can only cast Thunderous Smite, but you can't attack anymore- having expended all your attacks. You already attacked with your Hasted attack. You can't Two-weapon fight either, because Thunderous Smite is a Bonus Action.
TL;DR- Everything is legal, but you can't attack after using Thunderous Smite.
There are are several!
While the booming blade and green-flame blade cantrips are the most obvious spell-assisted melee weapon attack options, that type of enhancement isn't limited to melee attacks. The spells listed below revolve around ranged-only. There are some other spells that apply to both, but I've focused this on the ones that require ranged.
The following are some examples, but others exist if you look at the spell descriptions in the various source books.
Cantrips
Magic Stone (EEPC, 160) is the most obvious option here:
You touch one to three pebbles and imbue them with magic. You or someone else can make a ranged spell attack with one of the pebbles by throwing it or hurling it with a sling. If thrown, it has a range of 60 feet. If someone else attacks with the pebble, that attacker adds your spellcasting ability modifier, not the attacker’s, to the attack roll. On a hit, the target takes bludgeoning damage equal to 1d6 + your spellcasting ability modifier. Hit or miss, the spell then ends on the stone.
Levelled Spells
If you utilize your spell slots, you've got some other options. The Ranger spell list contains a lot of examples:
Lightning Arrow (PHB, 255):
The next time you make a ranged weapon attack during the spell’s duration, the weapon’s ammunition, or the weapon itself if it’s a thrown weapon, transforms into a bolt of lightning. Make the attack roll as normal. The target takes 4d8 lightning damage on a hit, or half as much damage on a miss, instead of the weapon’s normal damage.
Flame Arrows (EEPC, 156):
You touch a quiver containing arrows or bolts. When a target is hit by a ranged weapon attack using a piece of ammunition drawn from the quiver, the target takes an extra 1d6 fire damage. The spell’s magic ends on the piece of ammunition when it hits or misses, and the spell ends when twelve pieces of ammunition have been drawn from the quiver.
Best Answer
You need a weapon for all the smite spells (except wrathful smite)
Smite spells require a weapon
Thunderous smite says:
In fact, every smite spell has similar language specifically calling out the need for a weapon (except for wrathful smite see below). By a strict RAW reading though the weapon does not actually need to be the one that makes the attack, but you still need a weapon.
Regardless, I think it is pretty clear here that the spell is intended to utilize the weapon that you use to make the melee weapon attack even if the wording leaves a bit of a loophole. Either way, the spells do require a weapon.
The body/unarmed strikes are not weapons
The PHB says:
Suggesting that though body parts are not weapons, the unarmed strike still counts as a melee weapon attack but explicitly not a weapon.
And the Sage Advice Compendium clarifies this and makes it explicit:
Since the smite spells specifically call out a weapon in their description, that is what they require. And since unarmed strikes are not considered to be weapons, then they do not qualify.
Wrathful smite does not need a weapon
Wrathful smite is, for some reason, the only smite spell that does not specifically call out the need for a weapon.1
Thus, since the unarmed strikes do count as melee weapon attacks, a character would be able to use an unarmed strike to cause the effect, no weapon required at all.
1 - Thanks @Someone_Evil for catching this.