Understanding one another in the Fellowship

[Kuruharan]

You don't have to be taught to learn things. Besides, you can always teach yourself. And Gandalf is very old, so I assume he would have had more than enough time to figure out the language.

I'm afraid that you do have to be taught Khuzdul. It is not something you can pick up on your own. The reason is that (besides the Petty-Dwarves) they do not speak it in the presence of non-Dwarves. Place names and such being the only known exceptions. If they wanted to hold private conversations in the presence of outsiders they used their "gesture-speech" which was almost impossible for outsiders to see.

it might me implying that he [Gandalf] went through the painstaking task of learning the language himself and he does not need Gimli's help.

I doubt that he had the opportunity, or the mentor to teach him. He was a very busy wizard. I suppose that it could be argued that he learned it in the First or Second Ages when he was not necessarily in physical form. He could have drifted over to Middle-earth and spent a few hundred years studying the Dwarves to pick up their language. However, this is unlikely. For one thing, I can't remember that the Maia visited Middle-earth very much after the Dwarves woke up (might be wrong about that though, can't remember for sure). The other thing is, why would he want to spend the time? It would take a long while to pick up the language from this type of observation. Why would he waste his time doing something like this when there was little reason to do so?

And it never said he didn't know the language.

True, however the statement is dripping with that implication.