If the security certificate was sold/given to you by your host, it probably cannot be moved... but you should contact the certificate authority and ask them.

However, there's no need to pay for a security certificate. You can use Lets Encrypt [letsencrypt.org] which is free. I use this.

Ask your new host about free certificates before buying one. You may even wish to find a hosting company that supplies free certs, many do.

But leave everything as is at your old host until you are completely set up and running at the new host... then change DNS to point to the new server IP address... then after 72 hours, go back and delete the old copy of your site at the old host... and from then on only use your new host.

Your new host should offer instructions how to set up everything. If they don't, choose one that does.

if you paid for it then you should be able to move it and yes you can pretty much just take the private key and crt/pem files and copy them across. Ive done this on lots of dev/test servers using the same cert and just using a local host file on my pc to point to the domain and the cert still works / authorises the same even though the same cert is working on another server.

A certificate is a key pair, which is associated to a domain name. So you can use the cert with this domain anywhere. But, if your host is sneaky, they may not give you access to the private key file to prevent you from copying it elsewhere.