Wash gently with an unscented soap and warm water. If you want to be extra, put some alcohol on there. It’ll hurt like hell and a half, but it’ll clear out any bacterial. It isn’t practical to bandage every cut, so my rule is if you saw white, cover it. Pretty self explanatory. Avoid wearing super tight clothing and wash the area after exercise as sweat can irritate the skin. The most important thing to clean, though, is the actual blade itself. Always clean new blades and check for serrations (bumps/hooks on the edge) that could break off or catch in the skin and cause more damage than intended. Soak the blade in alcohol and store in a clean, dry place to prevent rust. DO NOT use rusty blades. For the love of god, don’t do that. Please.
I really hope whatever you’re going through gets better soon. Until then, stay safe.