The first step is going to be moving the site to a new server. I will do this over the weekend. I am pretty good at this and we should have very little down time. Since this means I have to change the DNS some of your ISPs might not get the updated settings right away.
Once the site is over on the new server and I know I have a good database backup then I will do the upgrade. Pretty much all in a 24-48 hour period.
If you don't know what all the technical jargo means, don't worry. You should not even notice the move, just the software