Honestly, don't listen to the nay-sayers and just dive into Ranked right away. Doesn't matter if you're not "good", what's the worse that could happen? You'd be Ranked Bronze, and there's no shame about it.
Now, about the real question of how many champs to main, 4-6 is already a lot, 2-3 is much safer/more efficient, and you really don't need more than that unless they're all super popular and never available.
Regarding your specific list: Ziggs and Velkoz are good ideas, similar so easier to learn together, but at the same time with different unique strengths (Velkoz is better against Tanks, Ziggs better to destroy turrets), and Anivia is difficult to master but with a high potential, so it's a nice champion to main. I'd suggest avoiding Azir, because he's a very "unstable" champion: lots of changes quite regularly, and can go in and out of meta super quickly, plus he just spent months in a row at an abysmal winrate. At least wait a couple months to see how the last rework settle down.
And for Sup, your 3 champs are great, but I'd suggest maining either Blitz or Rakan, not both, and use the last spot for a healer such as Sona/Nami/Soraka. That would give you more flexibility. But honestly, it's probably overkill for just a secondary role, you'd be fine with just Braum ;) (And even if Braum happens to be banned just the game where you're Support, you can always use one of your Mid Mages in a pinch. For real. Please just buy a gold item and a Sightstone.)
, 
, 
, and 
, but I'm not sure on who else; I don't really play assassins), and for my secondary role, support, I'm going to play 3-4 (
, 
, and
as of now). Is this a good amount of champions for me to learn? Thanks in advance, and any other champs you think I should try based on what I play would be great!
top and am going to learn 
jungle. As far as ADC goes, I might learn either 
or 
