On the PHB p. 72 it just says:

Improved Critical

Beginning when you choose this archetype at 3rd
level, your weapon attacks score a critical hit on a
roll of 19 or 20.

So my assumption is that on attacks made by Booming Blade or Divine Smite, the crit still applies to the regular attack's effects, but does the critical hit then effect with the Booming Blade's on hit Thunder damage, or the Smite's Radiant damage? What about with the bonus action smites? Like Thunderous Smite, does it affect those as well?

Basically, my main question is, does Improved Critical basically just boost your critical range, or does it specifically only improve the critical of regular weapon attacks from the attack action?

Improved Critical expands your critical range.

You normally only do a critical hit on a natural 20 roll. The Improved Critical ability changes this so a natural roll of 19 and 20 counts as a critical hit with all the benefits that this confers. (Doubling of damage dice, Auto hit, ect).

The only limit it adds to this extension is that it has to be an weapon attack. This means normal spell attack rolls do not get the benefit from improved critical. Important to note is that Spells that use normal weapon attacks to deliver their effect do benefit.

So then when you have an ability that does extra damage on your attack like smite then those dice are also double as normal.

It boosts your critical range.

As long as you make a weapon attack, you will make a critical hit on a 19 or a 20. In other words, you will double all damage dice if you roll a 19/20 on a weapon attack roll. From this question, a roll of 19 also makes you auto-hit your target.

This ability can trigger even on a spell, provided you are making a weapon attack as part of the spell. For Booming Blade, or Green-Flame Blade, you make a weapon attack.

As part of the action used to cast this spell, you must make a melee attack with a weapon

Also, Divine Smite can be enabled after you roll your dice, so you can decide to use it after you roll a 19, which doubles all your Divine Smite dice as well.