Never solo mid and try to get a laning partner who is a caster. This empowers your laning partner's spells by hitting the target due to Amumu's passive. Ryze, Annie, and Veigar are ideal because they all three have stuns, and while the foe is stunned, they're taking extra magic damage as a result of your passive, your Despair, and your laning partner's spells.
I FORGOT: Get the golem buff whenever you run out of it. lol. Also the lizard buff doesn't hurt and encourages players to stay out of your face because if they don't, they're getting magic resistance reduction.
Step 1. Take the Tank role, and STICK with the build for Tanking. If you stray from it, you'll just set yourself up for a quick death in teamfights.
Step 2. Starting Items: Doran's Shield, Health Potion x1.
Step 3. Ninja Tabi. Buy it in pieces if you need to, respectively in Boots of Speed, Cloth Armor, Ninja Tabi.
(It's the cheapest boots in the game that has the +2 movement buff and it's ideal for tanking for obvious reasons.)
Step 4. Know thy enemy. If 3/5 foes are mages, go Force of Nature > Banshee's Veil > Sunfire Cape x 2.
If 3/5 foes are physicarries, you can either go Sunfire Cape x 4, or Force of Naturex2 cuz it stacks, Thornmail to punish attackers for even attempting to widdle you down, and a Sunfire Cape to even further punish and help farm.
For summoner spells I go with Flash and Heal. It makes me outlast most hero 1v1 attacks with me killing them cuz my Despair's on. You can replace Flash with Cleanse if you're really weary of CC, because in most cases engaging with Bandage Toss into Curse of the Sad Mummy will get you stunned before you can set it off. In contrast, if they lack a snare or stun, you would go with flash to Bandage Toss in, Curse, Flash out, then live to harass them with your presence via Sunfire Cape and Despair. I personally end up dealing fcktons of damage to 5 people and barely making it out alive with Flash, then b out. It works probably 70% of the time.
Get Dodge, Magic Penetration, and Mana Regen runes, go for the Defensive tree and part utility for the mana regen and experience boost.
Here's what I do for abilities.
E, Q, W, E, W, R, taking R when you can and E above Q and W for the passive damage reduction, and maxing out Q after that for the harassment, then E because your initiator (Bandage Toss) isn't as important as the tools in your ability list that truly impact the way your win team fights, harass, or avoid ganks.
If you want more tips, add me. My Amumu's movin' on up, as my team says. :P
My fav tactic is running into a team with Bandage Toss, popping my ult, then attack each player once and watch their Health fall quicker than than mine does as they attempt to attack me instead of my teammates who are wailing on them. It's very frustrating and despite my constant damage output, they realize the fresh players in their faces just became the threat and I just walk away to go buy stuff. lol. Plus this build can take about 9 tower hits and live at Lv. 18. Possibly more. Tweak as you wish.
One last note: All this dodge stacking will let you dodge important projectiles including Parrley, Tristana's Ult, Teemo's Blinding Dart, and even yes - Turret shots.