Looking for a bit of advice! My site hosting company is currently also hosting my domain name (but fortunately they have put me as the registrant) but am looking at doing some serious upgrading. This is going to involve moving my site to a different host, having a back up host and naturally some domain name issues. Have read a little bit about hosting domain name separately and also having some third party DNS.. What is the safest and best way to go about this (or is it better having everything with the same hosting company?).

I have always recommended keeping domain registrations and hosting seperate, there are "some" hosts who take advantage of registering the domain for the client, in their name and then the owner loses control, look for a registrar that has a control panel, leaves the registrant and billing to you and doesnt require hosting from them

I will second the domain-hosting seperation. I've had a bad experience with it and so have many others. If the domain is in your name I would suggest transferring it to another registrar (or the same registrar into another account) that you control.