"Today, I am going to climb a mountain."
登ります (noborimasu) is an intransitive verb, which means it can't take a direct object. Direct objects are signified by を, which is why we say 山に登ります (yama ni noborimasu).
山に登ります (yama ni noborimasu) means to "climb a mountain" because as I said, intransitive verbs take に. If you use を, you change the nuance of the sentence.
When you use を with an intransitive verb, you are saying where you are doing something. 山を登ります (yama o noborimasu) isn't saying that you are climbing a mountain, it's more like saying you are climbing on a mountain.
Basically, what you said in your previous comment but the other way around.
No worries, you were absolutely right that を is possible and I neglected to address that in my answer.
What is で used for then? I thought that was used to signify WHERE things HAPPEN. Render me confused.
It's all related to transitive (can take a direct object) and intransitive (can't take a direct object) verbs.
学校で勉強します。(gakkou de benkyou shimasu)
I study at school.
食堂で食べます。 (shokudou de tabemasu)
I eat in the cafeteria.
で shows where the action happens.
部屋にあります。 (heya ni arimasu)
It's in the room.
東京に住んでいます。 (toukyou ni sunde imasu)
I live in Tokyo.
に shows where something exists / the target of the verb.
What we're talking about here, using を with intransitive verbs, is complicated and definitely an advanced grammar topic. Rather than saying it's showing where an action is happening, maybe you could say that it's showing the direction of the action? I haven't come across any good sites about this topic in English, so it's hard to express, but this is my feeling:
山を登ります。 (yama o noborimasu)
I climb all around on the mountain. The defined space of the mountain is the passage for climbing.
公園を散歩します。(kouen o sanpo shimasu)
I take a walk all around the park. The defined space of the park is the passage for walking.
Sorry if the meaning's not very clear, I tried my best...
I think it might also be one of those things that's grammatically correct but native speakers don't necessarily always follow. Considering how often we flout the rules of English grammar because we don't know/care what the proper form is, there's no reason to think the same doesn't happen in Japan. There are probably Japanese speakers who say "I think you meant to say 山'を'登る" just like you get people in English speaking countries who say "actually, I think you mean 'to whom,'" and "I don't know, can you go to the bathroom?" Shut up Janet, you knew exactly what I meant.
に is used slightly more often than を for climbing mountains. I would assume that the assumption is that you make your way through the mountain/mountain pass rather than going over the summit.
'Climbing in the mountains' would seem to be a better English translation.
If Duolingo did not accept を for this I believe it should be reported.
Edit: Feel free to downvote, but do your homework first.
From Living Language:
The Japanese comma is called とうてん tooten (formal) or てん ten (informal). Rules for the use of the Japanese comma are relatively loose. They may be placed wherever a natural break in the sentence might occur. An example of a common place you’d see a comma is right after a subordinate clause (such as a when clause and a while clause). You should keep in mind that the Japanese comma should not be placed immediately before a particle or conjunction.