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2 Questions that game up:

1. Can a squad with assault weapons that cause a wreck/explodes a transport such as a rhino be able to assault the squad that was in the rhino?

2. Can the opponent choose to move, shoot, AND Assault on the following turn as normal or are they unable to assault after their non-assault transport was wrecked/exploded from their enemy's turn?

Example: Top of turn 1, I blow up their rhino and berserkers come out, Bottom of turn 1, my opponent moves up, shoots, and assaults.

1, Yes <rulebook page 80 last paragraph before flyers section>
2. They cannot assault(unless it has some rule that allows them to)

Disembarking Restrictions
After disembarking, models can shoot, counting as having
moved, or choose to Run in their subsequent Shooting phase,
but cannot declare a charge in their subsequent Assault phase.

Did the Squad that caused the Transport to explode do so in the Assault phase? or was it from shooting attacks? If the Explosion was caused in the shooting phase then you can charge the unit that was being transported in the assault phase. On the other hand if the explosion was caused in the Assault phase then there is nothing more you can do to the transported unit.

The Answer to your second question is: Yes the surviving unit can act normally in the following turn as long as they are not pinned.

They act mostly normally but are prohibited from assaulting in their subsequent assault phase.

Also keep in mind that any shaken/stunned results may affect the passengers as well and impose other restrictions on what they can do.

 TheAvengingKnee wrote:
They act mostly normally but are prohibited from assaulting in their subsequent assault phase.

Also keep in mind that any shaken/stunned results may affect the passengers as well and impose other restrictions on what they can do.

So if I blow up their vehicle on my turn, they cannot assault on their turn?

Example: Top of turn 1, I blow up their rhino and berserkers come out, Bottom of turn 1, my opponent moves up, shoots, and cannot assault until their turn 2?

Yes. Unless it was an open-topped or Assault vehicle.

Just for reference, from the Rulebook FAQ (http://www.games-workshop.com/MEDIA_CustomProductCatalog/m3030001a_40K_RULEBOOK_v1.3_FEBRUARY13.pdf page 7):
Q: If a unit disembarks from a destroyed vehicle during the
enemy turn, can it Charge in the Assault phase of its own turn?
(p80)
A: No, unless the vehicle in question was an Assault Vehicle.