You can only liberate as a colony cities of yours that are on a separate continent from the one your capital is on. From what I can see, there has to be a minimum of two of your cities on that continent. So you can't liberate one-city island nations, unless you gift them to another colony you've set up.
If you meet those requirements, go to your Domestic Advisor and you'll see a new icon in the bottom right hand corner. Clicking this will allow you to choose which group of 2+ cities you want to liberate as a colony.
Once you've done that, you can then gift them other cities you also want to liberate. Be sure that in your initial trading with this colony, that you ask them to switch civics and religion to match yours. They might need to be given something such as your world map, in order to convince them, but generally they are quite agreeable about it.
After that, if you continue to treat them nicely, they will remain strongly loyal to you and can be very helpful in terms of resources, votes for you, sharing military struggles with you...etc
When you set up a colony, it's like your civ has had a baby. Congrats, you're a Daddy.
