First, you need to understand your playstyle. Each support champion or support-like champion caters to a different style. For example, you wouldn't go Lux support just to play passive in bot lane. Understand your playstyle, learn it. Then you need to adapt your style according to your adc/team.
Choose 3 very different kinds of champions to focus on and get better at in the support role. For Aggressive styles I suggest Morgana; it is really easy to land her Qs once you realize how huge her Q hit box is. For Passive styles, any champion with good peel will work such as Janna or Nami though Nami has a higher learning curve. For hybrid styles, any champion with good poke potential (this doesn't just mean damage) and peel will work such as Morgana, Lux or Zyra.
Once you get good enough, any support-like champion will work for any of the different playstyles. It is a matter of experience.
**Second, ** you need to understand laning mechanics in bot lane. As support, you play a crucial role in providing laning pressure and creating opportunities for your adc.
You can provide pressure based on any number of factors including your positioning, your poking habits, your specific champion and/or your item build. Lane pressure is all about psychological warfare. You make the enemy believe you are threat/reason enough for them to be wary and to back away as you approach. This provides your adc with safe farming and hopefully denies your enemy laners their farm and experience.
Third, to answer your specific questions, I suggest playing morgana as your first support champion. It is easy to play her since her skill set is relatively simple. She fulfills her role by landing her Q which has a large hitbox. Land her Q, then W, then if you have a good opportunity, go in and R. She is probably the most simple champion you can use.
As for higher Elo, the highest I ever got to was Plat II. So all I can say is, you need to be able to adapt. Be a "jack of all trades" as a support; this means you need to be able to play most support champions well enough. As your opponents get better in higher elo, you will understand that the nuances that didn't seem to make a difference in lower elo are now much more significant in higher elo. This is why you see people talking about certain laning mechanics that don't seem to matter/work in lower elo.
For learning the specifics of laning mechanics, I suggest reading up on it or looking up videos. Those will explain it best.