Here's the list of moves it can learn from leveling up:
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Ninetales and gulpin have like 20 moves each by leveling, starter pokemon also learn many moves by leveling A pokemon can learn better moves if you neglect evolving it for a while, vulpix can learn inferno at level 50 if you can hold out on giving it a fire stone until the . (Inferno is a great move)
Kecleon and Mightyena learn it while leveling up. Almost every other Pokemon can be taught the TM thief.
Pokemon will not learn moves while being raised in the daycare center. They will keep the same set of moves that you put them into the daycare with. However, moves that were missed can still be relearned through the move tutor.
Well, Metagross can learn all sorts of moves. I will tell you some of the TMs and HMs he/she can learn, and also some of the moves Metagross learns while he/she is leveling up. As Metagross levels up, he/she can learn[sorry,but these are not in order] Bullet punch, Psychic, Zen headbutt,Magnet rise,Take down, Confusion,Meteor mash and so on. In HMs and TMs, Metagross can learn Hyper beam,Psychic,Flash cannnon,Thunder[but I'm not positive about Thunder.],Earthquake,Secret power,Double team,Rock tomb,Shadow ball,Protect,Reflect,Sludge bomb,Cut,Rest,Flash,Return,Rock smash and Strength. There is many more moves, so I would suggest you go to Serebii.net Pokedex for all the moves. Hope this helped.
Egg moves are moves passed down from the father Pokemon when you breed Pokemon usually egg moves are TM's the baby if they can learn those moves will obtain them when they hatch out of the egg so whatever the father has TM wise or moves taught by a move tutor it will pass down to the child get it.