Current SRNA here. This is what I did.
1) Applied for ICU job. Did not ever mention anything about CRNA during interview. When asked where I see myself in five years, I replied with something like "Getting CCRN, precepting, working on my MSN"
2) Applied to CRNA school "in secret" never mentioning it to anyone except the two people on the unit I got a recommendation from. I did not use the unit manager for a recommendation since the manager never saw me working clinically (night shifter). Instead I used one charge nurse and my preceptor.
3) Went to CRNA school interview and was accepted. I did not tell anyone on the unit about this.
4) Charge nurse asked me one day after a few months if I ever got accepted so I told him yes.
5) Had to wait 6+ months between acceptance and the time to quit my job. Still kept everything as secret as possible. By now though the charge nurse must have told some other people because I had a few people asking me about CRNA school. If people asked me I would talk to them about it but otherwise I didn't initiate such conversations.
6) Once I knew I was going to leave the ICU in a month, I gave my manager 30 days notice. This was the first and only time I specifically mentioned the letters "CRNA" to my manager.
Some people might think I'm too calculating with my strategy but I took the approach that I thought was safe for me and as courteous as necessary to those who needed to know. Bottom line: Your manager doesn't need to know anything until you give your notice. If you were leaving for any other reason, you would just give your notice and be done with it.
There is no need to tell them 6 months in advance or give them a play by play for every interview and acceptance you get. If you want to be "friends" with your manager, that's fine but this is an extra, unnecessary step that can't possibly help you but could make your remaining time there more of a pain. Just apply, get accepted, and tell them when you are ready to leave the job (2 weeks or 30 days notice, whichever is the norm where you are).