It's important to note that Wyverns of the same elemental type cannot damage or be damaged by the same elemental breath attack, so for instance a fire wyvern is your best bet when alpha wyvern hunting, as it cannot be damaged by its fire breath.
There are more than one type of wyvern, they are fire, poison, lightning, and soon to be a Crystal Wyvern with the Crystal Isles free DLC coming, they are going to be a passive tame. The types of Crystal Wyverns are base, tropical, blood, and ember.
The wyverns spawn in wyvern dens, and can be found in the Highlands for Ragnarok and the Snow Desert in Valguero (I only do free DLCs).
You tame wuverns by going into the wyvern dens and finding nests with eggs in them. Once you have an egg, they need 10-15 standing torches, 4-5 fireplaces, or 20 (atleast) campfires. Until they are adults, they need Wyvern Milk to survive. That is all they eat until then.
Wyvern milk can be obtained by knocking out a female wyvern and collecting milk from there. Alpha wyverns drop more than normal wyverns.
Wyverns pick up larger dinos than other flyers.
Wyverns can breathe fire for fire wyvern, lightning for lighting wyvern, and poison balls for poison wyvern.