Upgrade and redesign of an existing website.

I have an existing website that needs to provide the same content but be completely redone. I have a search facility, login for my clients and need to update my own administration daily. I am looking to make the new site mobile friendly with good seo ability. CMS is very important to me.
The website address is www.kidsmanagement.co.uk.
I am open to suggestions as I only know of wordpress cms so something else maybe more preferable for this site.