The form -거나 is the basic form to give choices between two or more options. The form adds nothing of extra information itself on the type of choice or your feeling on those choices. It is simply a list of choices you want to talk about. As such, there is not a lot to talk about the form itself.
If you want to express a choice between two or more options, you can use the form -거나. -거나 is the absolute basic form to express choice in Korean. Therefore, it expresses little else than possible choices one can make. It does not add any feelings on said choices, nor does it add any information on the perceived quality of those choices. If you want to add feelings of info on the perception you need more advanced forms like -든지 -든지 or -느니.
가: 저녁에 집에 있거나 밖에 나가서 놀래요?
나: 저는 많이 피곤해니까 집에 그냥 있을래요.
A: Do you want to stay home for dinner or go out to play?
B: I’m very tired, so let’s stay home.
How To Use
Like -지만 and -고, the conjugation of the form -거나 is exceedingly easy. For this form, you needn’t worry over anything really. You just take the verb stem and attach the form -거나 to it. You even needn’t worry over the Korean irregular verbs. It is simply that easy. Do keep in mind that you can only use this form with verbs and adjectives, not nouns. To express a choice between nouns, you will need to use something else: the particle -(이)나. The particle has the same meaning, but it is exclusively used with nouns.
Lastly, like the form -고, sentences with the form -거나, generally, express the tense and such in the final verb of the sentence.