Doing WP site transfer from other hosting to another usually is quite simple and can be done in 30 minutes.
Generally, the process is to make a backup file of your WP site on your existing server, then import that backup file to your Cloud4WP hosting.
Here is the complete step-by-step process below.
NOTE: The steps below is only if your backup file is not too large ( less than 500MB). If your WP site size is larger than that then I suggest to do a site transfer using Duplicator plugin: https://wordpress.org/plugins/duplicator/
STEP 1: Make A Backup File
a.) Visit your WP admin dashboard
b.) Navigate to Plugins -> Add New
c.) Search for All In One WP Migration
d.) Install and activate that plugin
e.) Then visit the All In One WP Migration menu -> Export
f.) Make a file export (Export to -> File)
STEP 2: Update Your Domain Nameserver
a.) Visit your domain registrar (Godaddy, Namecheap, etc)
b.) Change the nameserver of your domain to your Cloud4WP nameserver
Then wait for few minutes, nameserver propagation can take up to 48 hours but usually it only takes 10-60 minutes to propagate.
After the domain nameserver propagates, continue to step 3 below.
STEP 3: Connect Domain Name to Your cPanel
If the domain name is used as the cPanel main domain, continue to step 4 below.
But if the domain name will be added as an Addon Domain:
a.) Log in to your Cloud4WP cPanel
b.) Visit Domain name -> Addon Domain
c.) Add the Domain name there as an Addon Domain
STEP 4: Transfer The Website
a.) From your cPanel, install WordPress to your domain name (I believe you already familiar in installing WordPress using Softaculous)
b.) Then log in to your WP site
c.) Install this two plugins:
d.) Then navigate to All In One WP Migration menu -> Import
e.) After that import the backup file from the Step 1 above
The website transfer should be successful by now. And you can log in using the same password and username that your WP site was using on the old server.