Felling smash, but unless you're playing an eldritch guardian getting the feats for that on a familiar is tricky. Similarly with ____ strike etc.
Share spells or deliver touch spells with something relevant - toppling spell + force punch or something.
A hyena spirit skin, if your familiar can use shoulder slot magic items, lets the wearer attempt a trip as a swift action. A legbreaker amulet of mighty fists might work if the familiar has a bludgeoning natural attack, and bites count as bludgeoning among other things.
Improved->Greater Trip can give you an AoO on the way down and when they use their move action to get back up. Give your familiar a Fortuitous weapon or Fortuitous Amulet of Mighty Fists, and they can do a second AoO whenever an AoO is provoked. If your familiar has Improved Unarmed Strike, you can give your familiar Vicious Stomp, which would pile on one more AoO on the way down after a Greater Trip.
Essentially, with Greater Trip, Combat Reflexes, Vicious Stomp and a Fortuitous Weapon/AoMF, it could have 3 AoO's after performing a successful Greater Trip, and then one more AoO if the creature tries to use a move action to get up.
creatures get these things :
trip ability (wolves etc have it) trip after a hit. grab (a LOT of grappler type have that) grapple with +4 after a hit.
both of the above abilities do not provoke.
but you said familiar. they don't get feats so it's really hard to get them this unless the normal creature already has the ability (i think the crab familiar has grab)
The Beast-bonded witch can give their familiar feats. The Eldrich Guardian (fighter archetype) can share combat feats. The Homunculist (alchemist archetype) can grant their homunculus familiar evolutions but I don't think they can take two point evolutions such as Trip.