Stress; as a native french speaker, we haven't got variable stress in prononciation, so this is difficult for me. Vocabulary acquisition is challenging too, as there are not as many obvious cognate with my native languages. And then there is the буква Х which my brain always interprets as the same as "R" in french, making any word with both "р" и "х" challenging. 

Of course, declension can also be difficult, but this is the sort of thing that can be remedied with more practice/exposure. 

Cases.

Transitive and intransitive verbs.

Sometimes it seems that for every English word their are five in Russian that are all slightly different and very specific.

Cases

The OP asked for moments, not facets. As with all languages, the most difficult moments are the first few hours of conversation for me.

As for grammar - 1) cases 2) verb aspect 3) verbs of motion

Stressed letters: where do we put the stress, why it changes and when? Also, the pronunciation of said letters, which changes when the stress changes.

I think I can manage cases, genders and the alphabet if I study a little harder.