Continue the usual sort of care you would give its current state, but ensure that it’s comfortable too. Some evolutions, particularly those for bug-type and reptilian pokémon, might be painful or otherwise uncomfortable, as their bodies begin to change and adjust before the full evolution occurs. Additionally, for most evolutions, there will be an adjustment period in which the pokémon will need a bit of time to get used to their new size or physiologies, so in general, it wouldn’t hurt to make them as comfortable as possible throughout the process. That means making sure they have plenty of easily accessible food and water for the days leading up to and through their evolution, as well as soft bedding and, in some cases, sturdy toys that can be clawed, chewed, or otherwise made to withstand destruction.
Also, this is something not a lot of trainers think about, strangely enough, but always do your research on your pokémon’s evolutionary stages. Furthermore, stock up on their needs. Evolved pokémon tend to need more food and space than their preevolutions, and in some rare cases, their needs may change completely. (You don’t take care of a huntail or gorebyss the exact same way you would a clamperl, for example.) Educating yourself and preparing beforehand will make the transition easier not only on your pokémon but also yourself.
Other than that, care is generally straightforward. Just be alert: your pokémon may be more excited or agitated immediately before or after evolution. Keeping a second pokémon at the ready to calm it down may help.
Best of luck!