How do I move my domain and email to google hosting?

I realise I might be playing a bit loose and fast with the terms so I'll try and explain.
I have a domain and and email hosted with a company (Hostek). Over the past couple of years, my email has gone down for no apparent reason. It's usually fixed within a couple of hours when I raise a ticket, but it's not a level of service I expect. Obvs google isn't going to be perfect, but I've never had my free gmail account go down.

Sooooo, what are the steps I need to go through and what can I expect my current ISP to do/provide for me to make the move?
Obvs I'll be asking them as well, but I can't help thinking I should have an independent view of what to expect.
TIA