Our tools vary from show to show, but it really saves time when you have a great character UI, a way to pick controllers of your character-- especially when you have a lot of work to do with fingers or facial features.
A good facial shape setup will go a long way. If you have to zoom and move around your character trying to find the right controller, then you're wasting time. So if there's not character picker, then I set up scripts (in Maya) that help me select all the fingers at once, or all the tail controllers, or wing controllers, etc.
It's also really helpful to have a script that will select all the keys that are part of your character in case you have to move the whole animation around. It is a huge time saver for me during my initial blocking passes.
Or, let's say you have to add some handheld motion to your camera and you need to set random keys for a long period of time, then it helps to have a script that does that for you. You just adjust the frequency and amplitude and go from there. If you had to do that by hand, it could take a long time.
Sometimes you need to zoom into your scene but you don't want to use another camera because you need that exact angle and setup of your shot. For this you can use a 2D camera zoom script. This helps me a lot when I need to focus on specific areas that are more detail oriented and smaller, like fingers vs. big body movement.
These tools actually exist online. You can find them on creativecrash.com in the Maya MEL script section, and I highly recommend them.
I also have a set of hotkeys within Maya that help me save time. For instance going back and forth in time, frame by frame, or toggling Nurbs controllers on and off, or other things that I use very frequently. The hotkeys are also laid out close together in a way that I don't have to move my hand around, just stretch my fingers a bit and that's it.
Of course there are also other tools for copying animation curves, mocap tools, cleanup tools, etc. that I use, but due to their proprietary nature I wouldn't be allowed to talk about them. The previously mentioned tools are all available online and let me have a pretty good setup at home and not just at work.