This lesson is about the several ways to say "or," but it is not exhaustive. Nevertheless, it will provide you what you need to know to understand how to express "or" correctly in Japanese.
か can also list noun phrases to mean "or." When you start a new sentence to list things with "or," you use それか. If you start a new sentence in listing with "or" but are making a question, you use それとも. か functions for both situations when you don't make a new sentence.
I intend to eat sushi or sashimi.
Grammar Note: か is not necessary after sashimi. Adding it would be old-fashioned.
Eat sushi, or eat French cuisine.
3. すしを食べる? それとも、フランス料理を食べるの？
Will you have sushi? Or, will you have French cuisine?
Will you write it by hand? Or, will you type it?
I'll cook. Or, I'll do the cleaning.
Practice: Translate the following. You may use a dictionary.
1. A nation like Japan or China.
2. I'll eat fish. Or, I'll eat pizza.
3. Will you go next month? Or, will you go next week?
4. He is around 20 years old.
5. Is he smart?
"Whether (or not)" is "か（どう）か". どう may only be added when the embedded question doesn't have a question word like 何 in it. When just か is used, the "not" is implied.
It is doubtful that he is his little brother.
Do you know if she is coming to the party or not?
I don't know whether or not I'll be able to finish my homework by tomorrow.
“Is tomorrow's weather going to be good?” “Uh, I don't know if it's going to be good weather or not”.
It is unknown whether the teacher went to the school.
I don't know who he is.
Grammar Note: The だ isn't necessary when followed by か inside a subordinate clause.
Variant Note: かいなか is a very formal variant of かどうか.
AかAないか: No More Than
This usage is exclusively used with counter phrases. As the examples below suggest, A is the same verbal expression in the affirmative and negative form respectively.
It costs no more than 2,000 yen.
The number of steps is no more than 1000.
He's no more than fifty years old.
または, rarely written as 又は, is used in situations such as when you want to express tolerance/allowance of either options presented or when out of two things you take one and get rid of the other.
To inform by either phone or telegram.
To either prepare meat or fish.
To put a 〇 to A or B.
Reminder Note: As for または, it gives the sense that either is fine.
もしくは, rarely written in 漢字 as 若しくは, is used in limited situations where you choose something out of several options.
Writing with a fountain pen or a ballpoint pen.
To contact via letter or phone.
Remember Note: もしくは should only be used if the options are not significantly different.
あるいは, rarely in 漢字 as 或（い）は, is used in situations where you are showing that things are alternate or both simultaneous, but it is not normally used in showing permission/allowance.
To either report the interview results by oral representation or essay form.
A capital city like Tokyo or Seoul.
Note: As for あるいは, the items must be of the same kind.
ないし, rarely spelled in 漢字 as 乃至, doesn't merely suggest A or B but A and B and what's in between. This is quite different from the other options. So, pay attention to this.
North or northeast
Etymology Note: As the characters show, this is not a combination of the negative ない and the particle し.
Reminder Note: All of these are rather formal and literary and would be replaced by か in the spoken language.