1. There's an animation canceling combo but you'll only get to use it in its current state for 2 weeks so don't bother.
2. Try to always have *either* deflect or dash off cooldown. Either of those abilities is a get out of jail free card for almost everything (including, notably, Roadhog's hook). The safest way to play is to keep both off cooldown until you come under fire you can't deal with then deflect for the full 2 second duration and dash to safety at the end of it. Playing like that you're very unlikely to die. Although you might want to use dash offensively if you want to kill medics.
3. BTW, you should kill medics. You're good at it and you're one of the few characters that can be mostly safe in the midst of the enemy. It might seem hard to hit a thin hitbox with your travel time weapon, but with practice even a jumping and dashing mercy will be an easy (well, easy-ish) target.
4. Double jump constantly. It might seem predictable, but the way Genji moves and the lack of large, easy to hit with splash damage makes Genji a difficult target even if you make no effort at all to be unpredictable. Like I mean just literally mash the jump key.
5. Genji's clip size can be annoying, so get used to working melee attacks in to extend it.
6. You'll just have to memorize which walls can be climbed and which roofs can be stood on. Its finicky.
7. You're main answer to your "hard counters" that can't be deflected (Mei, Winston, Zarya), is to climb walls or dash away, so always be ready with those. Climbing walls (until the nerf in two weeks) refreshes your double jump a single time, and then you can climb *each* separate wall surface once before your feet hit the ground. So you can basically fly when threatened if you know what you're doing. None of your "counters" have any realistic chance of following you into the sky, although Mei can pepper you with icicles on your way up.
8. For a new player I'd suggest to pretend the right clicks don't exist, especially due to point 1. For most players if you learn to lead your shots you can pretty much just spam at any range from sniper range to melee and the left clicks will do fine. Even if you have no hope at all of killing your target, just keep spamming, because...
9. Genji's ult is amazing. The idea is to constantly refresh your dash. Using the ult puts your dash off cooldown, so you can dash, ult, dash in. Then you want to pretend tanks don't exist unless they're your only target. You can kill them, but if you really want that team kill go for the squish characters. Two sword swipes (which is very rapid) will kill a squish character, 3 if they're being actively healed. So dash in, double hit, dash to the next target. Learn the length of the sword (practice, practice) and hit from exactly the edge of the range, and of course don't stop spamming that double jump key. Once you're done, dash right at the end to cancel the sheathing animation (if anyone is left alive, probably due to Zen's soon-to-be-nerfed ult). Oh and you can try and line up multiple squishy characters at once as if you were a Winston, but you do so much damage and have so much mobility that's a secondary concern. Long as you get kills, even 1 at a time, you can keep the train going. Oh and when ulting, hold onto deflect as long as possible, since its the main thing protecting you from barrage or death blossom.
To summarize: Dash, hit q (ideally from out of sight), dash in, left click twice, dash in, repeat last 2 steps as necessary, dash to cancel sheath. Spam that spacebar throughout, and save the deflect.
10. Due to the power of your ult and the fact that Zen's ult is the best answer to it, you might want to wait until Zen is dead or has wasted his ult to drop yours. Help him along by throwing things at him constantly.
11. If you have an Ana on your team, tell her when your ult is ready. Nanaboosted Genji ult is ridiculous. Although don't be too much of an asshole asking for it, there are other ways to use Nanaboost.