That depends on your own needs and preferences. We recommend that you do not get hosting unless it has 24/7 phone support.
99% of the time, nothing goes wrong. But when something does go wrong, it will always be at the most inconvenient time. It is best to have phone support which is available to you anytime.

We recommend it.
If you website is critical to your business, you can't always wait until business hours to get it fixed.

How to move my website?

If your old website is a collection of static files, then all you need to to is copy the files over to your new host.
That isn't usually how it happens, though.
If you are using a content management system (which you should be), then you will need to copy over your files, and also your database.
This can be done, but it can be very difficult.
Some hosting companies offer transfer support — helping to move your site onto their hosting. If you have a complex website, or you aren't comfortable with databases and config files, you'' want to find one that does this.