The animation does not survive
This is the thing to know before you start. A browser canvas holds one frame, so what you get is the first frame of the GIF as a still image. There is no partial success here and no setting that changes it.
If you need motion, the answer is an animated WebP or an MP4, produced by a tool that can read every frame — not by a page like this one.
Why the colours are already limited
GIF caps out at 256 colours per image. Any photograph inside a GIF has been through a palette reduction, and the banding you see is baked in. Converting to a format with millions of colours does not recover the originals; it just stops the limit applying from here on.
WebP is smaller than the PNG equivalent and keeps transparency, which makes it the best of the three for the web.
Where GIF still turns up
Old documentation, email signatures, forum avatars, and exports from software that has not changed since 2005. Converting the still frames out of them is usually part of tidying up an old asset library.
Questions
What happens to the animation?+
It is not carried over. This converts a single still frame — the first one — because JPG, PNG and WebP as produced here are single-image formats. If you need the animation, you need an animated WebP or a video, and neither can be produced by a browser canvas.
Why does the colour look different?+
GIF stores at most 256 colours. Converting does not add colours back, but it does stop the palette limiting anything further. Gradients that were banded in the GIF stay banded — the damage was done when the GIF was made.
Is transparency kept?+
Yes for PNG and WebP, which both have an alpha channel. No for JPG, which fills transparent areas with white. GIF transparency is one-bit — a pixel is either fully transparent or fully opaque — so there are no soft edges to preserve.