More and more, I find that getting someone's mindset to change is the major impediment to changing to a DevOps process. Once a person starts to believe in DevOps and accept a new way of working, they can rapidly start to improve the efficiency of their work. Moving on to change the mindset of a team (or teams) is even harder, but this is arguably the most important part of changing your software development process.
I don't have a great way of changing everyone's mindset, but if you want to start, I have a thought for you that might add a little work, but will help you understand how to make changes in the way you work and experiment with a better way to deploy database changes.
The tools are the easy part, and I want to help you get comfortable because the harder part is then trusting your process and helping others to learn how to trust a DevOps flow as well. To get you to start that journey, you need to learn one skill: how to execute your changes from the command line.
That's it. Think about how you will manually execute your tasks now, and then ensure there are command line execution processes for these. This could be a call to SQLCMD with a filename as a parameter or it could be a DACPAC deployment with sqlpackage.exe. If you are a Redgate customer, we have PoSh cmdlets for deploying changes with SQL Change Automation or command line SQL Compare calls.
A command line interface (CLI) is really the key for the DevOps toolset to succeed. That's only a small part of the process, since you need cultural change, you need to still model your database well and write good code, and you still need to learn how to improve your code over time. Adding a DevOps process allows you to spend time on code quality, testing, and skill improvement, instead of assembling changes and files for that next deployment.