Okay, I'm going to be blunt: The tutorial sucks. Hard. It doesn't even scratch the surface of the game and what you need to do to play it competently. Sometimes it even teaches you the wrong things. For example, if you go ahead of your wave even so much as a pixel, the tutorial admonishes "STAY BEHIND YOUR MINIONS!" Now that's fine if you're playing a ranged attacker like Ashe or Ryze, but what about Garen? That's who I was playing. We all know Garen's a tanky fighter, so we all know he can wade in ahead of the minions and take some hits. But whenever I got into melee range of a creep... "STAY BEHIND YOUR MINIONS!" I got so conditioned by "STAY BEHIND YOUR MINIONS" that I ended up not being able to play anything other than ADC for the longest time. And it's the same tutorial for all three champions being used. Nevermind that Ashe, Ryze, and Garen all have very different playstyles and roles that you can't just do a one-size-fits-all tutorial and expect players to do well at the game.
So here's my idea for a new tutorial:
Basic Tutorial Same tutorial as the howling abyss.
Fighter Tutorial with Garen Takes place at summoner's rift and explains the role of fighters in a MOBA, and fighter items. Also introduces the concept of split-pushing.
Tank Tutorial with Dr. Mundo Takes place at Summoner's rift and explains the role of Tanks in a MOBA and armor/MR items. Also introduces team-fighting and peeling.
Mage Tutorial with Ryze Takes place at Summoner's rift and explains the role of Mages in a MOBA and Magic Pen items. Also introduces comboing with abilities and usage of potions.
Assassin Training with Ahri Takes place at Summoner's rift and explains the role of Assassins in a MOBA and AP items.
Marksman Training with Ashe Takes place at Summoner's rift and explains the role of Marksmen in a MOBA and AD items. Also introduces the concepts of last-hitting and wave-control.
Jungle Training with Warwick, Master Yi, and Sejuani Takes place at Summoner's rift, and explains the differences between jungling playstyles, items, and smite. Also introduces the concepts of ganking and epic monsters with buffs (ie. Dragon and Baron.) Note that you should play this tutorial once with each champion.
Support Training with Alistar, Soraka, and Annie Takes place at Summoner's Rift and explains the differences between support types and corresponding gold-generating items and how they work. Also introduces the concept of warding, vision, map control, and smart-ping. Again, like the jungle tutorial, you should complete this one once with each champion for the full benefit.
Battle Tutorial with any of the above champions Ties all of the other tutorials together in an all-bot full game.
In addition, here are two enrichment tutorials for practice.
Last hitting and lane-freezing practice with any owned or free champion Taking place at howling abyss, you much get as much CS as you can laning against a random bot champion in 10 minutes. Also, no minion, allied or enemy, can be destroyed by a tower. If that happens, or you are killed by the enemy champion, the tutorial ends.
Vision control practice with any owned or free champion Taking place on summoner's rift, this is almost like "ward-ball." You have a sightstone and an unlimited supply of pink wards, as well as an upgraded sweeper lens. You can have up to 15 green wards and 5 pink wards on the map at any given time. You compete against an enemy bot champion to see who can get the most points by placing wards and clearing enemy wards in 10 minutes. 1 point is awarded for placing a green ward. 2 points are awarded for placing a pink ward. An extra bonus point is awarded for placing them in "optimal positions" (gank paths, jungle brushes, lane brushes for greens, if a pink reveals an enemy green, etc). 1 point is awarded for clearing an enemy pink ward; 2 points for revealing and clearing an enemy green. However, a point is deducted for every ward that the enemy bot finds and clears. Whoever has the most points at the end of practice wins.