Hi guys! Recently, I've been animating sprites, so I thought I might as well teach you!
STEP ONE. Make the frames for the animation. I'll be using the Dragonite and Abomasnow fusion frames I made:
Remember; Don't even THINK about making the files transparent. If you do, you could end up ruining the animation. I tried once, and this is what I got;
Also, Make all of the files the same size. It it involves wing opening and stretching the file size, then simply make the files at size of the wings when they open. Also, the maximum frames for the animation is 6, mainly because having 6 frames is easily to animate like this;
1 2 3 4 5 6 5 4 3 2
It goes back to frame 1 by default, so don't bother trying to add frame 1 again.
STEP 2. Let's find a good site! I highly recommend you use Gickr.com. We'll be using this one for this tutorial.
So you know how I said you would be animating 6 animation frames at the most? Guess what? If you count the number thing I did up at the top a little bit ago, then that in all is 10 frames.
The Maximum frame upload is 10 frames. SEE I'M SMART RIGHT RIGHT? *shot*
So really it's a win win.
Alright. Before we upload the frames, and click that very tempting to click continue button, we should do somethings so them the files don't turn into a clusterf**k
SIZE
If you ignore this, you will have something very very very bad happen. We wouldn't want that to happen!
All you have to do, is just put down the size of the files!
This file is 89x92 pixels. So we put that in.
SPEED
This is something that is a real pain in the ass. It can't go too fast, otherwise, you can't see the animation. You also can't make it go to slow, otherwise it will be rough. All I have to say is that you should try out every speed you can.
Also, there IS a custom speed button. The fastest you can make it go is 0.10.
Now click that Continue button that I know you have been wanting to press forever
STEP 3
*insert elevator waiting music here It helps =V
Alright! Now that it has finished loading, it's time for you to save it to your computer! (Unless it is incorrect in anyway, then you should fix it owo)

Personally I just use GIMP. All you really have to do is use each layer as a frame of animation (and thus it makes using transparency easy, so long as you don't combine the layers), and in the save as GIF dialog, there's an option for "save as animation." If you know what you're doing (okay fine it does take a while for you to know what you're doing with GIMP) then you can do, well, as much as a GIF can do.

Likewise. I use GIMP as well to string images together to form a GIF. In days gone by, I used Microsoft GIF Animator, which allows you to control every individual frame and its background settings. Combining that with GIMP usually settles any animation you need.