Updating WP server to newest versions of mysql & php

Hi,

We have a bunch of WP websites (only WP sites) on our server.
With moving a clients new site to this server we ran into some problems as the site was created on another server with higher mysql version running and this was not compatible with our servers version so we were missing some things.

We kind of want to update from Php 5.3.3 & MySQL 5.1.73 to newer versions.

We don’t think this would have a lot to the websites that are currently running on it, most are build within the last 12-18 months and don’t use any out of date plugins/themes. But some quick searches on the web do spawn some people with problems.

What would you all suggest, what should we be looking for etc.

we will obviously make backups etc. before we do this. but we would like some information beforehand on what to look for, as we don’t to have to much downtime on our clients websites.

Kind regards,

Martijn