Best champs to learn botlane with?

Thresh, Leona, and Morgana are probably the 3 need to knows for supporting, and it gives you a well rounded experience of a tank, mag, and utility support. but Braum is probably the easiest to learn and do well with and he has a little of everything. Janna and Sona are also good picks (though Sona may be nerfed off the face of the planet before too long)

I haven't done much ADCing but Lucian, Vayne and Cait seem to be among the favorites in that role

For support Thresh is a must, he's very well rounded and works with most things. For the other two, Janna and Braum would be my picks. You'd have a nice set of picks for various matchups.

ADC is a little more complicated. In general I would say Cait, Lucian, and Graves or Corki. But for ADC you'll mostly look to find the ones with kits and animations you can work with best. For instance I can play most of the ADCs well, but Ashe is by far my favorite to play. Just love her autoattack animation and kit.

Learn Draven. He is easy to use and difficult to master. If you get the hang of him and learn how to position yourself while working his mechanics, then ADC will come naturally to you.

For support, pick Taric. He has it all and will teach you when is the right time to engage on the enemy. Don't just randomly throw your CC. When you get the hang of him, switch to Blitz. He's a skillshot support which rewards you more heavily for landing his CC.