I would suggest waiting for 3.5.1 to be release before upgrading due to a number of bugs with 3.5.0, including 1 that may impact your upgrade of the MySQL database.
In general, you should be able to install the updated software and openfire will attempt to update your database automatically.
Right. I`ve done the same a couple of month ago. 3.4.5 (old server) to 3.5.0 (new/current server) and it is working like a charm. i had a mysql change from 4.x to 5.x but i noticed not even one error/change in the db tables. be aware of a collation change! this could be if you migrate db from one to another version.
Just backup everything an make a fresh install with 3.5.0 on the new server an then import all user and db tables.
If your server is open to the internet, you should update as soon as possible. 3.5.0 solved an DoS attack bug. However, you need to do a small change in the database scripts of 3.5.0, according to this thread.