Lol actually the two heroes I main 
[Amumu] - the first thing you need to know about amumu is aoe. He is an
aoe god.
Bandage toss - this allows you to lose the gaps, inniate in team
fights and make people fear being in range of you. I only get one rank
of this at level 2 until i’m forced to because your damage will come
from your other skills.
Dispair - this skill is amazing. Doing a percentage of the opponent’s
hp is godlike, it helps you destroy tanks, slay golems and lizards and
dragons like a beast, and make baron look like a chump. I level this a
bit faster than tantrum. Leveling it up faster makes getting golem and killing dragon easier. Also, when using this, if you are near a tower or just sometimes, its best to just walk next to the opponent while dispair is on and avoid hitting them, hitting them will actually stop your champ for a second and might allow the opponent to get away from you.
Tantrum - This is your basic nuke, as well as makes physical damage do less to you (minions/dragon/golem/baron kinda) It cools down faster each hit you take so if you can iniate in team fights and can get the first hit and force creeps to hit you it will allow you to spam it and wreak havoc on teams. Great for destroying creeps waves as well as just harrasment.
Ult (forgot name lol) - this is his AOE snare. One of the best ults in the game imo. Great for setting up team fights (annie,annivia,fiddle and any other aoe skills) You have to be careful with using this, using it right when you bandage toss into a team won’t help becuase they will already be snared, it’s best to wait a little bit in the team fight to get as many people as you can. In a 1v1 situation use it as they try to escape from you, doing it right will help you catch them for the kill.
Passive - reduces the MR of anyone you hit with your basic attack. In team fights just run right up to the carry or whoever you need to and hit them to reduce the MR. Doing this will make your other skills do more damage and just wreak havoc.
=Items=
My personal favorite build to go with him is as follows
Merc treads
Chalice of Harmony
Crystal heart
Sunfire cape x3
at the end of the game you can sell the chalice for another sunfire, Sunfire capes stack so having three of them will not only keep you alive and make any physical damage champs worthless, but it will also just bring the damage and help win team fights. Watch the HP go down.
Start the game with meki pendant and an HP pot, turn that into chalice asap and try to do golem after healing. Chalice allows you to keep dispair on all the time when you need to.
If the opponent is stacking alot of AP, you should still be fine. Merc treads and Chalice along with your masteries should help you out with that and not needing to deviate from that item set.
After gaining more gold, this is where the choice is up to you (and after boots). I actually switch it up from getting the first part of crystal heart which will give you the cooldowns that always help bring the pain, or starting with just a sunfire which will helps tons. Once you can start getting sunfire capes you will feel how much it changes and helps amumu.
Once you start rolling just get them sunfire capes and you win.
I personally don’t go ap because while it gives you HP, I myself like stacking armor and being on fire.
=Masteries=
I use a 9/21/0 mastery set putting points into ap, exhaust and going up until the magic pen and drop the rest into defense.
=Runes=
For my tank runes, I use HP reds, Dodge yellows, flat CD reduction blues and flat HP quints.
=Summoner spells=
I use exhaust and teleport myself. You can switch out the exhaust for ghost if you want, but those two work great for me.
=Spell leveling=
1 - E
2 - Q
3 - W
4 - W
5 - E
6 - R
and then i switch between leveling up tantrum and dispair getting the ult when I can. And bandage toss last.
[Ezreal] Ez is pretty much sagat. You will be firing your shots all the time, direct damage and going ap (my personal favorite build) a nice aoe burst with spell flux and his ultimate. Using his skill shot you can change the tide of team fights by making the other team start with less health as well as just snipe someone who has low life from across the map.
=Skills=
Mystic shot - this is your basic skill shot. It will hit the first target it hits as well as making your other skills cooldown faster. If you are building AD, this is the main skill you want to level faster because it is modified by damage and not AP. It will also apply any on hit effects to your target (lizard, black cleaver, lifesteal).
Spell flux - This skill shot will travel straight to where you shoot it and will not dissapear until it reaches its max range. It doesn’t affect creeps but it will do damage to any and all champions it travels over. If it hits your allies, it will boost thier attack speed. Great for a team fight to just throw it down the middle and hit several people.
Arcane shift - This is your mobility skill, it flashes you to wherever you click as well as shoots a blast to the nearest target. It can let you escape certain situations and is good for burst along with essence flux.
Trueshot barrage - This is your ultimate. It is a wide arc of magic that will travel across the map to wherever you shoot it to. Simlar to Ashe’s ultimate, but if this hits anyone it won’t dissapear. Great for team fights, ganks, sniping across the map or stopping that creepwave from killing your tower at bottom while you and the team are at top.
Passive - Your passive will increase your attack speed for every spell you succesfully land. For your ult and essence flux it will increase for EACH person you hit with it.
=Items=
My item buil
Ive played both AD ez and AP ez. I personally feel that AP is the stronger of the two and build him as such. The items I get are:
Sorc boots
Mejai’s
Void staff
lichbane
Archangel staff
zhonya’s
With ezreal the first thing I do is rush lichbane. By going flat AP your mystic shot won’t get the bonus damage from ap items, however with lichbane it will. Your mystic shot will apply the bonus damage to whoever it hits. This lets it increase in damage alongside your otherskills as you gain ap.
After getting sorc boots and lichbane, get a mejai’s. If you are playing right you should get stacks of that pretty good from ganking, or just starting a teamfight with your ult or just a single essence flux.
I start with doran’s ring and a pot
=Masteries=
I got 9/0/21 for ezreal. Putting 9 in offense up to magic pen, and the rest into utility. Putting it into utility helps with the mana regen that he needs.
=Runes=
I use a very similar runes set to my tanking ones, with magic pen runes in the red instaed of flat HP
=Summoner spells=
I use exhaust and teleport. Teleport lets you get to a team fight asap and help with a gank if needed, and exhause helps your surviavbility when someone like Yi or tryn or whever tries to kill you.
=Skill leveling=
1 - Q
2 - W
3 - E
4 - W
5 - W
6 - R
continue to put points into essence flux and level up arcane shift and mystic shot as you feel necessary.