Learn Finnish: Minä katselen televisiota ja olen rento. - I am watching television and I am relaxed.

Katselen comes from the verb katsella, which is formed by adding the frequentative suffix -ella to the basic verb katsoa (to watch/look). This suffix gives the action a more continuous, casual, or leisurely feel. While katson televisiota simply means "I watch television", katselen televisiota emphasizes that you are casually or continuously viewing it over a period of time, which perfectly matches the relaxed mood of the sentence.

Why does televisio have the -ta ending here?

The word is in the partitive case (televisiota) because watching is an ongoing, uncompleted action. Verbs of looking and watching, like katsella and katsoa, usually take a partitive object because you are engaged in the process of viewing, rather than completing a final action that "finishes" the TV.

Could I use the verb rentoutua (to relax) instead of olen rento?

Yes, but the grammar would change. Rento is an adjective meaning "relaxed" or "laid-back", so you use it with the verb olla (to be) to say olen rento (I am relaxed). If you wanted to use the verb rentoutua, you would conjugate it as an action you are actively doing: Minä katselen televisiota ja rentoudun (I am watching television and relaxing).

Do I have to include the word at the start of the sentence?