How do I go about adult adoption when one is in their 20's and has no family? What requirements/rules should be considered during the adoption process?

Are your parents okay with this? How old are you?

Adoption doesn't work that way. You can't register with an adoption agency. You are placed with an agency when and if you become an orphan or your parents abandon you,

I'm sorry you feel this way. However, adoption doesn't work that way. It's not your choice, but a choice your parents or children's services make.

It depends on the country. In Japan, young men are often adopted even after they are adult, when a family with a business doesn't have a son to take over the business.

If you are a young person, in 20's, you find someone that wants to adopt you.
There is no agency that I know of. Normally this would be a family you know well, or a person with a close relationship. And they would adopt you.

The area of the world, where you live. Has to allow it.