This is my quide on amumu who is my favorite for jungling and winning games, I've used many builds and say this is by far the strongest ive used yet and if you follow the build and skill order im sure you'll have an easy time ganking and racking up those kills early in order to get off to a good start. This is my first guide and I hope this helps you:)

When jungling amumu I would grab flash so I can catch my enemy or get out of a sticky situation and pick up smite to clear your jungle faster.

For jungling tantrum is a great skill point to use first because of the reduced physical damage you take and when activated attacks surrounding minions or enemies with magic damage, secondly I go with Despair to clear through the jungle even easier and painlessly, despair damages surrounding units each second, then I go with bandage toss at level 3 when im ready to start ganking to make it easier.

I go with x9 Mark of insight for the magic damage, x9 mark of resilience for the armor, x9 Greater glyph of shielding for magic resistence and x3 greater quintessence of potency for even more powerful magic damage

In team fights or for engaging one you always want to be the first one in and for greater chances of winning team fights try to hit your ultimate as soon as possible helps team fights tremendously and most of the time you always want to go for the squishies but standing in the team fight spamming tantrum is a huge help.

Start with armor and 5 pots for jungling makes it a lot easier because of all the time you can spend in the jungle early game and not need to recall. Get a philo stone and heart of gold for sustain early on and I usually go with mercury treads for the extra magic resistance in which some cases you should go with but I find the enhanced mobility to be extremely useful for the fact that catching up to you enemies and staying in front of them is a priority for amumu and ganking is fairly easier, if they do have a lot of magic damage I recommend mercury treads and the sunfire cape is a positive for reduced physical dmg and that it damages surrounding enemies. I get banshee's veil for the magic resistence which helps a lot if I go with boots of mobility