Learn Swedish: Jag sätter på radion.

You are right! The verb sätter usually means "puts" or "places". However, when paired with på, it creates a phrasal verb (partikelverb) meaning "turns on" or "switches on". You can think of it as "putting on" the power. Another common way to say this is slår på.

Can I split the words and say "Jag sätter radion på" like we do in English?

No, you must keep them together here. Unlike English, where you can say "I turn the radio on", Swedish phrasal verbs keep the particle (på) right after the verb (sätter) and before the object (radion).

Which word should I stress when saying sätter på out loud?

In Swedish phrasal verbs, the stress almost always falls on the short little particle word, not the main verb. So you should put the emphasis heavily on på, pronouncing the sentence like "Jag sätter PÅ radion".

Why does the word radio have an -n at the end here?

radion is the definite form, meaning "the radio". Because en radio is an en-word that ends in a vowel, it just takes an -n at the end to become definite. Just like in English, Swedish requires the definite form here for a specific object in the house.