Looking for hosting company
The top two complaints we hear the most from organizations about their previous providers are:


lack of responsiveness/customer service to inquiries.
consistency in maintenance and updates.


When we take over hosting and maintenance services, we try to identify any technical debt that exists so it can be addressed as early as possible in the engagement. 

I recommend paying attention to what sort of site/code audits the provider does upfront. It will likely save you in the long-term and start everyone off in the relationship with better and more information.