Upgrading an old WordPress install to a new one on a new host

Most of the time any changes that need to be done on upgrade are database related. Core files can be freely swapped pretty much.

WP keeps track of this by comparing db_version option (stored in database) against the current value (from core files) on admin load. If there is mismatch it runs the upgrade routine to perform all the necessary changes, until the version is up to current.

In your case the easiest way would probably be:

1. Unpack latest stable WP core at new location
2. Copy old database to new location (or simply reuse old one) and configure new core to see it
3. Copy content folder to new location (configure new core to see it if not placing inside the core folder)
4. New core would pick it up as existing WP site with outdated database and perform the necessary updgrades