Before you can even think about upgrading your system from one release to the next, please take note of the following:
You must be currently using a system that is version 0.62 or above, as all releases prior to version 0.63 do not come with database upgrade scripts. If you are using a version before 0.62 then your only option is to make a clean install, removing your current data from the system.
The upgrade is carried out in two parts. Both parts must be done before the system is usable again, and so only carrying out one part of the upgrade will lead to an inconsistent system. This will require downtime for you system, and your user base needs to be aware of this downtime. The upgrade itself should take no more than a couple of minutes, but during that time, noone must have access to the system.
Ugrade scripts for the database are written to upgrade from one specific version to the next. If you are trying to upgrade from say v0.62 to v0.7, missing out v0.63 (or any other versions that might have been released in between) your upgrade will fail and result in an unusable system. In order to upgrade through a number of versions, you must sequentially execute each individual version increase (in the right order and not skipping any versions) from your current version to the required version.
Upgrading the PHP scripts is as simple as installing the new version of scripts on the server in place of the old ones. Just follow the instructions for PHP scripts.