What film should I use in a Fujifilm Clear Shot 20 Auto, and can I develop it myself?

There are 2 questions here - you really should consider them separately.

What film to use... The camera uses 35mm film, which, even still, is not all that hard to find, certainly online. But there are some supply chain issues these days, so prices/stocks have been negatively affected.

You should look for the following:

• new film, not expired film
• 35mm, colour, C-41, ISO 400 or ISO 800
• as cheap as you can find

C-41 means it can be easily developed by a lab, and ISO 400 or 800 film will be better able to handle all kinds of lighting with this camera.

Highly recommended - find and read the manual for the camera. I would not consider this camera a "vintage" camera. Yes, it's a couple of decades old, but that doesn't imply any classic quality. It's a cheap compact camera that didn't even autofocus as far as I can tell. It might be a bit of fun to use, sure, but I wouldn't recommend it for someone who wants to properly get into film photography. Try to use it predominantly outdoors in sunlight - indoors you won't get good results with this camera, even with the flash.

How to develop film yourself... That's a complicated process. I would start off simple - try a couple of rolls of 35mm colour film and pay for a lab to develop and print/scan the film first. See if you like it first before going down the route of developing your own film.