Thresh is a jack-of-all trades sort of support with many different tools in his kit. A shield, an escape tool, engage, disengage. Band-Aid is the word that comes to mind. If you can't get a hold of a champion that specializes in something, you can pick him and still be servicable at anything a support needs to do. Of course, he still does have his own weaknesses, and we'll be looking at them to see how to take him down.
Passive – Damnation
Thresh can harvest the souls of enemies that die near him, permanently granting him Armor and Ability Power.
Q – Death Sentence
Thresh binds an enemy in chains and pulls them toward him. Activating this ability a second time pulls Thresh to the enemy.
W – Dark Passage
Thresh throws out a lantern that shields nearby allied Champions from damage. Allies can click the lantern to dash to Thresh.
E – Flay
Thresh's attacks wind up, dealing more damage the longer he waits between attacks. When activated, Thresh sweeps his chain, knocking all enemies hit in the direction of the blow.
R – The Box
A prison of walls that slow and deal damage if broken.
Thresh aims to continuously engage on their opponents. If you don't let him engage on you, part of his kit goes right out the window.
Standing on Threshs lantern can prevent people from taking it while you kill them.
Save any dashes or gap closers for after he uses his flay, as it causes a slow and he can peel you away with it if you cannot dash to him/your target.
There is value in making any moves after Thresh has used and missed his Death Sentence, since he no longer has a way to hard CC anyone that is a priority target.
Your plants can block his hook and you can snare him out of any engage, which just makes life hard for him. Be careful not to get caught though, you're still very squishy.
You engage way better. If he engages on you and your AD Carry, you can counter engage, you'll win most of the times, the extra damage from your passive winning you the trade.
You can just deny all of his CC and move on with your life. If he takes his hook in, you can Q him and you'll still be CC immune, leaving him sitting there with nothing to show for his efforts.
Mikaels and Quicksilver Sash have value against Thresh, as do all manner of armor penetration. You can buy boots of swiftness if you're concerned about not dodging his skillshots, but more importantly are wards, so that you don't get caught from a skillshot from the fog of war.