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A nice little free program (that runs ads to support it) is 2020 [hotfreeware.com], which does screen captures, resizes and converts file formats, and does a fair bit with creating and editing images for being cost-free.
There are a lot of free programs out there, I imagine there must be some listed at ODP.
For banners and graphic logos, look at a bunch of different companies' websites, and commercial ad banners and make notes about the ones that catch your eye... Why do you like them, what got your attention, would you actually click on the banner, etc. Once you have a good idea of what kind of look you're going for, almost any free/shareware graphics program would allow you to actually create the graphic.