"I will climb the mountain, then go to the river."
Not in this case. The て form of のぼる here is indicating a sequence of events. て is not just a sentence joiner~
As 3d9y2 has said, the primary function of the ～て form is to join two verbs or two phrases to represent a relationship or a sequence. ～て does not mean a command by itself - ～てください is. It shows the verb before て and ください has a relationship (requesting the other party to do something).
You can also use て form as a command though, it's just more direct and less formal than saying てください, 見て！= look 食べて！ = Eat! It's still softer than the imperative form though, afaik.
It is important to note, by appearance, ～て form means a command, but actually not. It is because ください, which means "please could you give/do me," is omitted. (Much like わたしは is omitted from most of the sentences, or ありがとう which omits the ございます that follows for a complete sentence.)
The Te form has a lot of applications, really takes a textbook to learn all of them. That's why it's just called the Te form rather than by an English equivalent like imperative.
After seeing this question multiple times, I've finally decided to post an answer. The て ending is not the imperative. The imperative forms are <sub>なさい、</sub>え、～ろう、and ～よう。 Get out! = 出て行け！出て行きなさい！ Eat! = 食べよう！食べなさい etc. ～て indicates incompletion. It means there's more to come. It is used to join the first verb to another verb for sort of a "compound verb" like 出て行く or 変えてくる or 返してください。(Notice the last one is the "polite command" you were talking about, which is an oxymoron in Japan. It's a request, not a command because imperatives are rude by nature.) It is also used to connect clauses together.
山を登る and 山に登る are both correct but with a different meaning.
山に登る - mountain as a destination.
山を登る - mountain as a pass-through point. e.g. 山を頂上（ちょうじょう）まで登る I climb to the top of the mountain.
Grammatically, It actually could be either, but which one you use depends on your purpose for going.
It's true that に can be and is used for more specificity, emphasizing the destination (and purpose in having gone), while へ is a more generalizing particle, indicating emphasis on the journey to a destination, not necessarily with a purpose in mind.
Also, however, any へ can be replaced with に, but not every に can be replaced with へ.
The greatest weakness to Duolingo is that it doesn't give you specific direction as to what is most naturally spoken by a native to the language..
Not really. へ Is more "in that direction." Like 川へ行きます is saying "I'm going in the direction of the river, and I might get distracted and do something else" whereas 川に行きます is saying "I'm going directly to the river, and not anywhere else"
well the distinction is subtle so I would saay they are generally interchangeable.
I have not heard a case in this [location] + [movement verb] setting that one can use に or へ but not the other. Really it originated differently, but nowadays people tend to use it without distinction.
There will be cases that you can only use に or へ but that is definitely not the case for 行きます／来ます
- 南へ向かいます (this can also use に but I will definitely say へ is a better choice)
- 駅の前に止まります (this cannot use へ)
まして is allowed but is too formal in common use cases. It is certainly common in formal letters.