Announce the spell,
identify the mode and the targets,
use mana abilities and pay for the spell.
The correct procedure is in the rules.
I actually *know* how to cast a spell... There are procedures, and there are real life ways of playing. You don't say "Pass priority" ten times per turn in a real game as you have to on MTGOnline, for example. There are shortcuts and acceptable plays. Some rules are written, right. But rules are just guidelines. In your first situation, It's pretty clear that the player tapped mana with the intention of casting his spell, so backing up includes untapping lands.
Your Tendrils situation is pretty different.