For Competitive: Meteor Mash as it is a STAB move that ends up with 150 base power. You might want to replace Hammer Arm for Earthquake though. But both Hammer Arm and Earthquake are pretty equal on Metagross and what you want to use just depends on what you want to hit. Zen Headbutt can also be replaced by Earthquake.
Ingame you'll probably want the set of Ice Punch, Zen Headbutt, Earthquake, and Meteor Mash. Hammer Arm will lower your speed, which is not so desirable as Metagross will otherwise outspeed the foe if it is at a high enough level. This is not a problem in competitive, as Metagross is on the slower end of things.
For against the Elite 4: Meteor Mash, Zen Headbutt, Earthquake, and Thunderpunch
Metoer Mash is powerful enough to take down any foe that is hit neutral by it. So most of the Pokemon of the Elite 4. So that is a must.
You'll want Thunderpunch, as it helps against Will's Slowbro, as well as Lance's Gyarados and Charizard. Earthquake to deal with Lucario from Bruno and Karen's Houndoom, as well as Will's Bronzong (which has the Heatproof ability). For the final move you'll want Zen Headbutt for a strong hit against Koga and Bruno.