Can someone explain when to say spróbować and when postarać się?
"spróbować" is more like simple "to try", and "postarać się" like "to put an effort". Here they both work well.
'Tego' instead of 'to' will not work here because of the cause, right?
Yes. Here, the verb that acts on "it" is "zrobić". Which takes Accusative, and as it's some general 'it', it's "to".
If it was just "We will try it", it would be "Spróbujemy tego", as "spróbować" takes Genitive.
Why can't the answer here be "spróbujemy to robić"? Instead of "zrobić". There is no indication in "We will try to do it" that the action will be completed.
I believe that robić can't be used because it's present tense. If spróbujemy is in future tense, then so must be the verb that specifies that action.
I think it's fine. "We will try to keep doing it this way from now on", makes sense to me. Added.