I want a support like Janna that works into everything and for everyone. However, I don't want to play Janna since she's often taken. What's a good generalist support I can learn (prefer tankier ones) without hard counters.
Personally, I think that Nautilus is probably the best option for a generic tank support.
His ultimate and Q are great for picks and initiations, his E and W provide a deceptive amount of damage in fights and his kit is loaded with CC.
Braum is pretty good. Just put your shield up when it's needed, land your Q, and in teamfights make sure you melee everyone so your team can proc stuns on them.
Alistar is a good tank and easy to use as well (and free). It's difficult to mess up your Heabutt > Pulverize combo now, and he can take a significant beating when his ult's up.
Karma. She ends up pretty tanky with supp items and brings a butt ton of easy to use utility, plus she's a monster early lane.
If you just want a simple, easy support that works in most teamcomps, I highly recommend Soraka. However, if tankiness is an absolute necessity, then try Taric, Tahm Kench, Blitzcrank or Thresh.
tl;dr play blitzcrank
GENERALIST support, you say? 
I would also suggest 
.
Tank supports that are good: and maybe even 
. Though I would say 2 really good supports (non-tanks) that you can play would be 
