Hanami

Is hanami for free or do I have to book a ticket for it?

Depends on where you intend to watch hanami, but generally it is free. However, some places, like enclosed gardens and parks, may charge an entrance fee. More here on some popular hanami spots with info on fees:

http://www.japan-guide.com/e/e2011_where.html

"Hanami" is just cherry blossom viewing. If you are enjoying cherry blossoms, you can say you are doing/enjoying "hanami".

Hanami can be done at many places for free, such as riversides, parks, etc.

But other places may or may not have a fee.
Temples typically charge fees for entry as they do year-round. Most shrines are free but some may charge during hanami season. Many castles only charge to enter the castle keep, but there are also castles that charge to enter the castle grounds, so you have to look up which castle you will visit to know.