Learn Russian: Хорошая погода радует людей.

Радоваться means "to be glad" or "to rejoice" (the subject feels the emotion). Радовать (without -ся) is a standard transitive verb meaning "to please" or "to bring joy to" (the subject causes the emotion). Here, the weather is causing the happiness, so we use the non-reflexive form.

Does радовать take the same case as радоваться?

No. The reflexive радоваться takes the Dative case for the thing you are happy about. But the non-reflexive радовать in this sentence takes a direct object in the Accusative case, because the weather is doing the action directly to the people (людей).

Why is the word for "people" written as людей here?

Людей is the Accusative plural of люди (people). Remember that for animate nouns (living things) in the plural, the Accusative case always takes the exact same form as the Genitive case.

How would I express this same idea using the reflexive verb радоваться from this lesson?

You would need to flip the sentence so the people are the subject (feeling the joy) and the weather is the Dative object (the cause). It would become: Люди радуются хорошей погоде. (People are happy about the good weather).