You can also use Intimidate to weaken an opponent’s resolve in combat. To do so, make an Intimidate check opposed by the target’s modified level check (see above). If you win, the target becomes shaken for 1 round.
For almost all purposes, there is no relevance to the end of a round or the beginning of a round. A round can be a segment of game time starting with the first character to act and ending with the last, but it usually means a span of time from one round to the same initiative count in the next round. Effects that last a certain number of rounds end just before the same initiative count that they began on.
Fear effects are cumulative. A shaken character who is made shaken again becomes frightened, and a shaken character who is made frightened becomes panicked instead. A frightened character who is made shaken or frightened becomes panicked instead.
My question is this, if he moves first in the turn causing the creature to become shaken for one round, and the next round uses a different fear ability does the enemy become frightened(aka fear stack)?
Or would the fear effect from the intimidate check have worn off as it would expire at the start of his turn.
Alternatively if he succeeds on intimidate and before the start of his next turn an ally used a fear effect on the creature and makes it frightened does the effect move back down to shaken after the turn is over?
Yes, the shaken fear effect would end at the start of your next turn, reducing the state to what it would be without the Intimidate effect.