Avoid Information Overload
There is a lot to talk about with Magic. Start simple with the turn phases, spending mana, casting spells, card types, sorceries, and instants. Basically cover all the things the instructional cards provided mention. It's a good way to start with those free sample decks that are handed out by most stores. So don't explain all the possible options yet what you can do in your draw step for example. For now stick with drawing a card.
If a card specifically does something special, like an upkeep trigger, of course explain it then. Gradually add more options the player can do (depending on the cards they have and draw) and explain how the choices influence the game. Do you cast in your precombat main phase or postcombat? And explain why that could be (more) efficient.
Let Them Make Their Own Decisions
… and show them afterward how different things could turn out. It's okay to let them make a "bad" choices. That way you can show the new player the implications of a different line of play. Think of a scenario where instant speed is of great importance or maybe a timing example with an activated ability. Show both situations and check if it makes sense to them. Also take your time with this and ask the player if they get the point of what you're trying to explain.
Make It Visual
My favorite example of this is explaining flying. Pick up your creature with flying and fly over the blockers without reach or flying and hit the D20 die. Such a demonstration helps understanding and recalling what flying means.
Still, not everything is that easy to visualize (how on earth would you tackle proliferate?!), but it makes explaining Magic a lot more fun for the both of you. Luckily the basic cards that you're working with are also not that complicated. Summoning sickness is also a nice one to show, and with that, haste as well. The game almost comes to life with this approach and that's awesome.
Repeat Explanations or Rules As Long As It Takes
Things like the stack, the turn structure, combat damage, power and toughness, or the difference between +X/+X until end of turn and +1/+1 counters can sometimes be confusing, I've noticed. Keep repeating those rules or ask them to explain to you what they think it means. Correct them in a positive way and don't be frustrated if you need to explain something multiple times. What I've heard back in my college years, when I needed to learn, is that you recall things after hearing them seven times. At one point, the new player will remember the rule!
Another example of something that I repeat a lot:
"But my 2/2 can attack your 1/1 right?"
"No, you always attack me. I choose if I block or not."
I had to explain this over and over again! Also note the lack of the planeswalker option to attack. Not too much information at first!
Don't Give Them Too Many Cards at Once
I believe customization of decks to be one of the most fun parts of Magic. But it can also be overwhelming to a new player. So many choices … I recommend not buying a booster box at once. Just start with a few boosters or donate a pile of cards of your own bulk collection. Let them know their cards and hopefully create a certain craving for new and better cards!
From there, show what kind of products are available step by step, and help with opening boosters if needed. You can use this moment as well to explain about rarity, foils, and maybe even things like masterpieces or similar. It depends on the person how fast this goes. If the game speaks to them, they'll be hooked and get their own boosters as soon as possible!
Explore the Colors
… and try to find the right one or right color pair for them. Before playing with all the rules and such, it is useful to discuss the colors, their philosophy, and how they are represented. Green is all about growth and life, where black is corruption and power at all costs for example. You know the drill.
Keep it short and simple, because as you know, every color has so many exceptions. Maybe you know the other person pretty well and can suggest one or two colors already. But ask away, let them tell you what colors they "feel." It's also no problem if they don't know. Just start and play the game! They'll know what feels right and what kind of affinity they have.
Magic Is About Winning the Game … Eventually
Don't hold back, but do it in a positive way. Letting the other person win is not a good way to introduce Magic. The drive to win and become a better player is important. Play like you would normally play and should you win, no problem! Explain that Magic is a lot of losing at the start—at least for most people. It takes time for someone's first own victory, and they should be able to own it. This is especially true when new players start building decks themselves, where there is a lot to learn about mana curve, amount of lands, and correct mix of creature/noncreature cards.
What I always like to do is to emphasize the moment where the game changed in either player's favor. Explain what happened and why the game shifted to a win or loss. More often than not, it leads to an insightful moment for your student. They were so close at winning the game! So keep it relaxed, but aim for the win as well. But leave your tier one deck at home. You might demonstrate the varying power level of some decks, but don't match an intro deck against a Legacy deck!
One last extra piece of advice: play the first two or three games with open hands and guide your new player with the options they have. That way you can track what's happening and help the new player in following the full game.
With these tips, I hope you can become a great tutor for Magic: the Gathering! Do you have tips to add? Write them down in the comments below. I'm curious about your experiences.
P.S.: Bonus points for you, if you know where the title reference is from!
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