I try to upload an image to my product at my fresh magento installation, but I can't find any way how to do this.
If you don't want to enable Flash Player in your browser, you can install Magento Patch SUPEE-8788 too.
According to your Magento versions, you should download the exact version of patch file. Otherwise the patching is likely to fail. And of course, you should apply other patches too beforehead.
Release Note: https://magento.com/security/patches/supee-8788