Moving a website can take 30 minutes while another website could take 2 to 3 hours or a few days.
It depends on the size of the website, whether it has a database, what size the database and how many files need to be transferred over to the new server from the other.
I moved a simple wordpress blog from one web host to another web host and it only took 30 minutes to complete the move.
That was moving the files and the database.
For a larger forum I run when I moved it from shared hosting to cloud hosting it took me all day since the database was huge and it took awhile to download it to my computer and then upload the database back to the new server.
Then I had to import the database using ssh instead of through phpmyadmin since it was too large.
Moving a small database under 50 MB in size is simple but moving a large database that's over 200 MB is a much more involved process.
The cloud server I host my large forum on is un managed and for advanced users that can work the command line which is not easy for a beginner.
So not only did I have to move the website myself I also had to install a control panel and php, mysql and patch and secure the server and I still have to keep the server up to date myself.
Some web hosts will move the website over to their servers for you free of charge if you ask them too.
If you don't know how to move the website yourself then get someone to do it for you.
You also need to update your domains name servers to point to the new server and the propagation can take 24 to 48 hours so be sure to leave your website live on the old host for at minimum of 48 hours so your site will be accessible.
If it's a forum just set the forum on your old server to read only so that no new posts will be added and put up a message that the forum is being moved to a new server and all access will be restored once you're on the new server.