The IC Tahiti is a wonderful property with a beautiful view of Moorea. I wouldn't be in a hurry to rush out on your second day. Have a leisurely morning and relax - you're on vacation.
If you go to Moorea first thing, your room will not be ready. I stay at the Hitlon, and every time the lobby is full of people who took the red-eye flight from LAX. They arrive on Moorea at 8:00 a.m. and sit in the lobby, looking tired, waiting until their room is ready in the afternoon. (I don't know why they don't change their clothes and spend the morning at the pool on those comfy loungers.)
I've done both. I slightly prefer the ferry. It's very nice to get out on the water after a long flight. The ferry cost a lot less than the flight (see exception below).
One time I took the flight from Moorea to Tahiti. I bought a "Bora Bora Pass" from Air Tahiti
https://www.airtahiti.com/multiisland-fares because it only cost 10 euros more that a Tahiti-Bora round trip. So I was basically getting the Moorea-Tahiti flight for 10 euros. That flight is very short (15 minutes in the air). It is nice to see Moorea and Tahiti from an airplane.
Yes, I got a welcoming gift and a fruit plate and bottled water. Don't expect a room upgrade. I've never gotten one. Some others have. It's not a sure thing.