A big part of it is just going to be just learning the fights. Know when the adds are going to pop, and where, and do what you can to be there. If you know adds pop when the boss is at 70%, stop doing GCD stuff at 71-73% (depends on how fast it's going down) and get your target switching fingers ready. If it's a fight where they pop in the same place, and it's safe to drag the boss there, do it.

Also, don't panic, prioritize. This also comes with knowing the fights. Which adds are the most dangerous? How fast are they going to die? If it seems like chaos, take a breath and go one GCD at a time, since that's about all you can do anyway.

With most fights, the adds fall into one of four categories. 1) You need to pick this up NOW. 2) You need to pick this up before it causes too much damage to someone else. 3) You probably can just ignore this. 4) You are SUPPOSED to ignore this.

The only ones you really need to freak out about are category 1, and those pretty much always spawn at a predictable time and in a predictable place. And you won't run into them until 50. Use the time until then to practice calmness and prioritization.