Well, let's say "исследование".

In short:
"Расследование" is for "investigation", about criminal or special cases. For example: aircraft crash is subject for "расследование".
"Исследование" (originally - exploring of landscape) is for "research" in scientific or professional way.
"Изучение" - has shift to "studying/learning" meaning.

In case of 5 it is a little bit strange for me to use these nouns for such activity. Sounds a little bit as comic exaggeration. I think something like "мне надо начать выяснять куда лучше поехать на летние каникулы" is more natural. Note, that "выяснение" as noun will be strange also.