Travel on Shikoku

I was wondering if we need to hire a car for the whole trip around the island. We are thinking of starting at Tokoshima and then onto the Iya Valley, around the Coast back up to Uijima and onto Matsuyama. Is there a rail around the island. I am assuming we will need a car in the Iya Valley. Also is it a busy tourist time. Would we need to pre book accommodartion. We are hoping to stay traditional Japanese, even though we realise now it will cost more than we initially thought. Really looking forward to a peaceful tour where old culture can be experienced.

The following map shows all intercity rail lines on Shikoku:
http://www.japan-guide.com/list/e1107.html

There is no rail or efficient public transport in the Iya Valley. A rental car is recommended:
http://www.japan-guide.com/e/e2024.html

Also is it a busy tourist time.

You did not mention when you will be traveling!

Would we need to pre book accommodartion.

I recommend to make advance booking at all times. Just dropping by a hotel is unusual (except for business hotels) and a potential cause for stress and time loss.