The only way to improve your ability to handle large volumes of traffic is to handle large volumes of traffic. That's not quite the tautology that it seems to be. Just like anything else, only encountering it and pushing through will get you there.
Obviously, that's not the answer you're looking for, so here's a few things to consider:
Prioritize based on the urgency of the need. Meaning, if you have someone LUAW with an inbound, keeping an eye on the exiting aircraft and getting that guy off the ground before the inbound has to go around is more important than someone receiving a pushback command.
When I'm controlling a busy airport like KLAX or KJFK, I make sure everyone on Tower and in need of sequencing or clearance is taken care of before I go through a sweep of pushback requests, etc. I don't bounce to ground for every pushback request instantly. If they wait a few seconds, there's no danger. If I don't clear or sequence those in the immediate airspace, that's more of an issue than a guy having to sit for a sec at the gate.
In the air, if it's busy, don't be afraid to assert your control over the airfield. If you have pilots atop each other, you can separate with a 360 (not on final, that's go around). You can utilize extend downwind. But mostly, make sure you sequence as early as you can ascertain which will be the correct sequence. If pilots don't follow sequencing, don't be afraid to just let them come on down then send them on a go around.
Many controllers on TS tend to try to call everyone's base, etc. There's no need for that, typically. Let the sequencing be your control. If they don't follow, send them around.
Work your way out from tower to the end of your airspace, more immediate need to less so. 25 miles out, give them a pattern entry, then you have some time while they continue inbound to work on ground, etc.
Monitor speeds and spacing while planes are invound. Know the plane you have at the hold short for departure. Is it an A388? Or a 319? That affects how much space you need to get them off the ground. Have a plan for what you're gonna do when the inbound lands before it lands. Do you have enough space to land up? Do you need to send the second inbound around?
This probably doesn't come out in the written word as I think about it in my head, but there really is no substitute for experience. You can't find the ability to manage huge volumes in a book. You just have to be stranded at KJFK and work through it. Much like building up speed on a guitar, pushing your processing speed can only be done with practice and experience.
Of course, I'm talking about what happens when you become IFATC (where you'll gradually build up from low- to high-trafficked fields) on Expert.
On the Training Server, you just wait for the guys to take off from the ramp and skip the line :)
[Edit: The above should say SOUP :). Sorry, I like to write.]