How to make a transparent or animated GIF
Paint by Zaweba's Export Animation writes GIFs with its own optimiser: median-cut palettes, dithering, frame differencing and transparency handling.

Both kinds of GIF come out of Export Animation in Paint by Zaweba. Build your frames on the Timeline, then choose GIF, and Paint's own writer builds the palette, handles transparency and squeezes the file down with frame differencing and an optional lossy threshold. A single frame with a transparent background gives you a transparent still GIF the same way.
GIF is an old format with tight limits: at most 256 colours per frame and only fully see-through or fully solid pixels. Getting a good looking, small GIF is mostly about working with those limits, and the optimiser's settings are there for exactly that.
Making the animation
Open View > Timeline. You can record frames from layer visibility, opacity, position and Layer Art, set a delay for each frame, choose how many times it loops, and use tween to fill in the frames between two states. A logo that fades in, a button that pulses or a sticker that bounces can all be made this way from a few layers.
For drawn animation, Layer > Animation gives you animated layers with cels and onion skins. Either way, the Timeline plays it back so you can check the timing before export.
Export settings that make the difference
In Export Animation choose GIF. The palette can be shared across all frames or worked out per frame, using median-cut to pick the best colours. A shared palette keeps colours steady from frame to frame; a per-frame palette suits animations where the colours change a lot.
Dither scatters pixels of the available colours to fake the ones in between, which helps gradients and photos. You choose the kind of dither and its amount. Less dither means smaller files and cleaner flat colour; more dither means smoother shading.
Frame differencing only stores the pixels that change from one frame to the next, which can cut the size of a mostly still animation dramatically. The lossy threshold lets very small changes count as no change, so even more is skipped. Raise it bit by bit and watch for shimmering in areas that should be still.
Getting transparency right
Because GIF pixels are either fully see-through or not at all, soft edges are a problem. An anti-aliased edge that blends into transparent ends up as a ring of half-coloured pixels. Plan for the background the GIF will sit on: if it is going on a white web page, a faint pale fringe won't show. Clean, hard edged artwork gives the neatest transparent GIFs.
Keep the background layer hidden or deleted before exporting, so the empty areas really are empty.
When GIF isn't the best choice
The same Export Animation also writes animated WebP and PNG sequences, and WebM or MP4 when the suite's FFmpeg is installed. WebP handles soft transparency and many more colours at smaller sizes, and video formats are far better for anything photographic. Use GIF where you know it will be accepted everywhere, such as email signatures and older forums.
Paint by Zaweba is free software that runs offline, so you can make and fine-tune GIFs without uploading anything to an online converter.