I have been playing Amumu actively since early Season 9 and have been playing League of Legends since Season 1 so I have a lot of knowledge about the champion and its best matchups. I hit Diamond 2 last season with a 65% win rate and with Masters 200lp MMR when I was grinding ranked.

When playing amumu you should not be looking to fight until at least level 3, most of the time playing amumu your goal is to stay alive until level 6. However, amumu can make an impact before that depending on jungle matchups and how the lane states are, so dont be afraid to look for ganks before level 6. Once you have ult you want to be ganking lanes or looking for objectives every time your ult is up. With your 2 Q's and Ultimate most fights are winnable as long as you hit your abilities. When your ultimate is on CD you can still look for fights just be more careful about enemy whereabouts.

After the red smite nerf i take blue smite in almost every matchup. red smite is still good into strong duelists and tankier opponents like bruisers. however i like blue smite for the slow/mov speed buff, and the extra burst to one shot squishies with your combo.

Playing amumu you aren't the strongest 1v1 champ however with all the cc he has you are a lot stronger than some people realize and you can take advantage of that. So when you are deciding what side of the map to play towards early game, first you must look at the enemy jungler and try to get info on where they are starting. If it is a jungler you cannot fight early game such as Warwick, you must path opposite of them and drop a ward to make sure you are not getting vertical jungled. The best thing to do vs an early game dominant jungler is to just stay alive and play towards the opposite side of the map they are. If you are against a jungler that you can take early then you can decide which side of the map you want to play off early, as well, if the champion isn't too strong you can also look for an invade because Amumu can actually 100-0 quite a few champs in the early game. Just be careful and don't overstep because you can be cleaned up easily from being out of abilities.

With amumu best thing to do is repeat gank your strong side because amumu isnt the most mobile champion so you cannot be everywhere on the map and help every lane like some junglers can such as Warwick, Hecarim, Rammus etc. So use your heavy cc to steamroll your way to end the game. Usually this will be through bot/mid because with Amumu's AOE ultimate it puts a lot of pressure on dragons which allow you to get soul pretty easily if you focus on it. As well as Amumu's ult being AOE it's best used bot side where there will be multiple enemies instead of using it only on one enemy like mid or top where you can just use your Q's to CC them.

Amumu has pretty good clear unless if you fall behind. just make sure you are pathing efficiently and not taking coinflip ganks when you have guaranteed gold on the map when your camps are up. If you coinflip a gank and die the enemy jungler will benefit off your camps that you left for them. However, of course if you have a good dive/countergank setup make sure to not lose the game over a wraith. Once you get Liandrys amumu clears camps pretty quickly and can use his Q to go over walls to get there faster. Another tip is that amumu's W does %HP meaning he shreds baron and dragon insanely fast and can solo baron with 3 items.

Best thing to know when playing amumu is how to chain cc. First make sure your W is active before you engage because otherwise you will miss out on a bit of damage forgetting to turn it on mid fight. First you will land your q, auto, e, then ultimate, auto twice, then q again and most champs will be 100-0% from that combo because of amumu's %hp damage. Also now that amumu has 2 Q charges don't think you always have to land your q on a champion. When ganking you can q to a minion to get closer then Q to your target making it very hard for them to be out of range.

I am a firm believer that Hybrid amumu is the best version of the champion as compared to full AP/full tank. However, if you arent as comfortable on amumu dont be afraid to build him as a tank. Hybrid AP/tank amumu allows you to still be very tanky while being able to burn the enemy and 100-0 a lot of enemy champions. As well as being able to kill Epic monsters very quickly, securing soul for your team.