There should also be training programs in it too:
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CS Training Program: Setup a time limit then try to get as much CS as possible within that time limit.
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There can be a farming under turret mode.
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There can be a farming with or without ally minions mode.
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There can be a last hit farming mode (hitting them once they reach a certain threshold of health based on your AD).
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Jungle Clear Program: Farm the Jungle to test out builds, routes, counter-jungling, soloing dragon/baron, etc.
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There can be a clear time mode, where you try to clear faster and faster or try out which routes are faster.
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There can be a training wheels mode for new junglers that walk them through when and where to go normally and when to smite.
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Skill-shot Training Program: A mode where you hit targets with your abilities, not only does it show you the range and speed but also exactly what they do.
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There could be Beginner, Intermediate, and Advanced modes where beginner would be simple hitting targets at different distances, intermediate would be hitting moving targets and randomly spawning targets, advanced would be targets with high mobility abilities like Lee Sin or Ahri.
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Item Build Program: Simple mode that shows you all the different items and what they do exactly.
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It could show you which items are good in which situations and what are most effective against certain builds and enemies.
Why do i think this is a good idea and not just ez mode get gud casul much? I think there are too many new players not correctly learning the game because the training mode is highly lacking. There are also smurfs (mostly of those whose accounts are permabanned) which are making the game for new players highly unenjoyable, i see way too many posts about them quitting because they dont understand the game and no one will help them. Well personally i think sticking them against Bots is a terrible idea, you can build pretty much any build and still beat Bots once you learn the game a little bit. Doesn't mean you can play with real people though. A more expanded and advanced tutorial is needed now, lets face it the community is only getting more and more toxic, permabanned accounts make new ones and act 5x more toxic. There is no solution to them other then remove new players from them until they can learn how to play well enough to ignore them.
So honestly i think Riot should really really consider making an advanced tutorial, i wish i had one when i first started. I am too scarred from my first time jungling that i hate doing it, basically i was terrible and everyone hated me. This happened a few times and now to this day i hate doing it, i dont know what to do and no one helps me all that much. If i ask for help or advice its always "/all 'Oh geez gg noob jung ff20'" and then i do something wrong and they act surprised. This also happened when i tried ADC, so i try not to ADC and when i do i make sure i let everyone know. Its not really healthy and i really wish there was a better way to practice without other people.
Well sorry for the long comment, i just want to say that i agree with your Sandbox idea but i think it should come with a Training Mode with advanced training modes for all the roles, specially with the new jungle with all it has to offer.