Yes, to begin your domain transfer you will need to “unlock” your domain if it is locked (usually it is by default). Login into your registrar’s site and there should be an option to lock/unlock somewhere. This will allow the transfer to initiate.
From here, depending on your TLD you will need either a Transfer code/key or to verify via email. For dot com/net domain names, you will need to obtain a transfer code from your current registrar as well. This is fairly new but it should be an option inside your registrar’s page as well. For dot ca/us and most other domain names you will need to make sure you have access to the email address that appears as the administrative contact in the whois information of that domain. This is to approve the transfer when it is put through.
Once this is out of the way, you are ready to order a domain transfer from another registrar / hosting company.
The 48 hour time period that @Dog mentioned is often referred to as the DNS propagation time and is merely a heads up about the length of time it could take for name servers to update their information (if they have it already cached). From experience (I own a web hosting and registration company) you’ll be rocking out within a few hours tops and isn’t something you should be too worried about.
In order to avoid any “downtime” make sure you copy your existing site over to a new hosting provider before you transfer away. This includes all your html/php/asp files, your databases and your email. Your current host should be willing to help, or at least point you in the right direction so that your new host will be able to do this all for you (find one that will). Do try to wait a couple days before telling your old host that you will not be renewing, to avoid them shutting things off before hand (they shouldn’t since you likely paid until the end of the month/year though)
This way when you transfer the domain and the name servers are updating, the files will be accessible on both servers and no downtime will occur.
Let me know if you need any more information on registration or hosting, I’d be happy to help.