A creature with the universal monster ability grab deals the natural attack damage of the natural attack used to start the grapple automatically only when the creature both begins its turn already grappling a foe and makes a successful grapple check. That damage is automatic no matter what the creature opts to do after having made the successful grapple check, be that move, damage, or pin. (Likewise, meeting similar conditions usually allows the creature to employ attacks tied to the universal monster ability rake, too.)

A creature with the universal monster ability constrict deals its constrict damage upon making a successful grapple check—including the combat maneuver check that initiates the grapple!

Hence a creature with both the universal monster abilities grab and constrict that successfully grabs its foe has already dealt the foe its natural attack damage (from having bitten (or whatever) the foe to use the grab) then deals its constrict damage on the first turn. On the second turn, if the grapple's maintained, the creature automatically deals its natural attack damage and automatically deals its constrict damage and can attempt to move, damage, or pin (and, possibly, rake).

You can read more on ENWorld from 2010 and on the Paizo messageboards from 2009 and 2013.

Below are three scenarios that demonstrate these abilities in action. I've chosen as the protagonist the really, really poorly optimized level 7 iconic fighter for ease of use. One hopes PCs do better. The average party level of Abel the fighter is 6. The tigers Bob and Cecil are both challenge rating 4, while Don the anaconda is challenge rating 5. All three encounters should be easy for Bob. (Hint: They're not. Never adventure alone!)

Abel the fighter fights Bob the tiger

Abel the fighter is hired by a small village to defend it from a tiger that's been eating the villagers. Abel is a terrible choice for this duty as he has a Perception skill bonus of +0, but the villagers are desperate. Bob entered the village from the other side and snacked on a villager, but the villager's screams alerted even the usually oblivious Abel. Abel draws his weapons and rounds the corner to see Bob 30 ft. away. Initiative is rolled. Bob's lucky and beats Abel's result.

Round 1

Bob takes a full round action to make a charge, designating Abel as the victim. As part of the charge action, Bob moves 30 ft. Bob makes an attack with his claw against the flat-footed Abel. The attack hits and deals damage. Abel is dealt 11 points of damage. Bob opts to use the universal monster ability grab associated with his claw to attempt to start a grapple. Bob succeeds. Both gain the condition grappled with Bob controlling the grapple. Bob's turn ends.¹

Abel takes a full action to make a full attack against Bob. Using his off-hand weapon would require both hands, so he decides to stick with just the longsword. The grappled condition imposes a −2 penalty on attack rolls, yet both Abel and Bob suffer a −4 penalty to Dexterity, evening things out for Abel. Abel easily hits and deals damage with both attacks. (Bob's AC is, like, 10 at this point because of the −4 penalty to Dexterity and the −2 penalty to AC due to the charge action.) Abel deals Bob 27 points of damage. Bob has 18 hp remaining of his original 45 hp, while Abel has 59 hp remaining of his original 70 hp.

Round 2

Bob takes a standard action to make a successful grapple check, his CMB +5 beating Abel's CMD and Bob's claw—with which he started the grapple due to the universal monster ability grab—deals Abel 10 points of damage automatically and Bob opts to deal Abel damage with his bite. Abel is dealt another 13 points of damage. Bob also makes 2 more claw attacks due to the universal monster ability rake. Both claw attacks miss.² Abel has 36 hp remaining of his original 70 hp.

Like last turn, Abel takes a full action to make a full attack against Bob using his longsword. Despite Bob's AC being a whopping 12 now, Abel still hits with both attacks. The second attack kills Bob the man-eating tiger.

Abel the fighter fights Cecil the cunning tiger

A few weeks later, Cecil the tiger—a much more cautious beast than his cousin Bob—realizes Bob's not defending his hunting grounds anymore. All those yummy villagers! Cecil's been lurking and pouncing on villagers as they went about their chores rather than invading the village and eating them like dumb ol' Bob. Cecil is cunning. Abel has once again been hired by the village to defend it. Cecil knows instinctively the swordsman must be eliminated if he's going to snack on any more villagers. Cecil gets the drop on imperceptive Abel. Initiative is rolled. Cecil wins.

Surprise Round

Cecil takes a full action to make a charge, designating Bob as his target. Cecil makes two claw attacks and a bite attack, opting not to grab with either successful claw attack and opting instead to grab with his successful bite attack. Cecil deals Abel 34 points of damage and uses the universal monster ability grab to start a grapple with Abel. Cecil beats Abel's CMD and both gain the condition grappled. Abel has 36 hp remaining of his original 70 hp.

Round 1

Cecil takes a standard action to maintain the grapple and beats Abel's CMD. Cecil automatically deals 13 points of damage to Abel as Cecil via the universal monster ability grab as Cecil started the grapple with his bite. Then Cecil opts to deal his bite damage again to Abel for another 13 points of damage. Again both of Cecil's rake claw attacks miss. Abel has 10 hp remaining of his original 70.

Now, Abel's been here before, and he won, so he does what he always does: he takes a full action to make a full attack with his longsword, easily hitting Cecil twice for 27 points of damage. Cecil has 18 hp remaining of his original 45.

Round 2

Cecil takes a standard action to maintain the grapple and beats Abel's CMD. Cecil automatically deals 13 points of damage to Abel as Cecil via the universal monster ability grab as Cecil started the grapple with his bite. Abel is dying. Then Cecil opts to deal his bite damage again to Abel for another 13 points of damage. Abel has been reduced to below his Constitution in negative hp and Abel dies.

Abel the fighter fights Don the anaconda

Abel's walking through the jungle where Don lives. Don's hungry enough that even a metal dude looks like a meal. Don waits until Abel walks under the tree in which Don lurks and declares his intention to attack Abel. There's no freakin' way Abel notices Don, so initiative is rolled, and, while Abel wins, Don's acting in the surprise round.

Surprise Round

Don takes a free action to fall deliberately 10 ft. from the tree he's in (with his Acrobatics skill modifier of +15 he's fine) and takes a standard action to attack Abel. Abel is flat-footed. Don's attack hits and deals Abel 14 points of damage. Don opts to employ the universal monster ability grab tied to his bite and attempts to start a grapple. Don needs a 13 or higher and succeeds. Don deals able another 14 points of damage due to Don's universal monster ability constrict. Abel has 42 hp remaining of his original 70 hp.

Round 1

Abel's been here before: He takes a full round action to make a full attack against Don, easily hitting Don twice for 27 points of damage. Don has 32 hp remaining of his original 59 hp.

Don takes a standard action to make a grapple check to maintain the grapple. He, fortunately, succeeds. Don automatically deals his bite damage (the natural attack used to start the grapple) and automatically deals his constrict damage and opts to deal his bite damage again instead of moving Abel or going for the pin. All told, Don deals Abel 44 points of damage over the course of the three events. Abel is at −3 hp and dying. Don eats him.

¹ Bob could have made a bite attack then a claw attack then another and final claw attack and opted to grab and make the free grapple check using that last claw attack, but Bob's a freakin' tiger. He's used to winning encounters by grappling.
² Attacks gained from rake still suffer the −2 penalty for the attacker using them in a grapple. The adjusted attack bonus for the claw rakes is then +6. Abel's AC is 22.

No, your turn does not end

Part 1: Full Attacks & Grab

A perfect example for this is the Dire Tiger, which has 3 attacks, all of which have the Grab ability.

So you're a dire tiger, and you make a full attack. You hit with your bite, choose to grab, and succeed on your check. You now have the Grappled condition, which states:

A grappled creature takes a –2 penalty on all attack rolls and combat maneuver checks, except those made to grapple or escape a grapple.

It does not say anywhere in the Grab special ability that your turn immediately ends. Additionally, in the Grapple section it says:

You can take any action that doesn’t require two hands to perform, such as cast a spell or make an attack or full attack with a light or one-handed weapon against any creature within your reach

Specifically, this is referring to the creature being grappled, because the grappler is unable to take a full attack having used their standard action to maintain the grapple. However, in the case of Grab no standard action was used, and the grappler has the same Grappled condition that the grapplee has.

Because there is nothing in Grapple or Grab that specifically prevents a grappler from making a full attack (aside from a standard action that Grab ignores), and it is specifically stated that someone with the Grappled condition can take a full attack, you as a Dire Tiger may continue to take attacks after your Grab at a -2 penalty.

Part 2: Greater Grapple

Yes, if you have not used your move action for the turn, you can immediately roll to damage the grappled creature as a move action.

Once you have grappled a creature, maintaining the grapple is a move action. This feat allows you to make two grapple checks each round (to move, harm, or pin your opponent)

Part 3: Rake

You're still a dire tiger, and you've remembered you have the Rake ability.

A monster with the rake ability gains two free claw attacks that it can use only against a grappled foe

It does not say you rake based on a maintain check, it only says you get the claw attacks to use against the grappled foe. No, you do not get to rake twice in the same round (you can only use the two claw attacks once).

Edit: Additionally, with Greater Grapple and Rake you still cannot use Rake on the first turn, because Rake states:

A monster with the rake ability must begin its turn already grappling to use its rake—it can’t begin a grapple and rake in the same turn.

Part 4: Constrict

If you were instead an Octopus with 8 grab attacks and constrict, you could take advantage of this line:

You can, as the creature that initiated the grapple, release the grapple as a free action

This would allow you to Grab, Constrict, release, and repeat for each grab attack you possess. However, keep in mind:

Free actions don’t take any time at all, though there may be limits to the number of free actions you can perform in a turn.

Your GM could still limit how many times you can release the grapple.

Edit: Again with Greater Grapple, Constrict states:

A creature with this special attack can crush an opponent, dealing bludgeoning damage, when it makes a successful grapple check

If you chose to hold the grapple until the next round (and possessed Greater Grapple) you could make two grapple checks and do extra constrict damage with both.