As you get acclimated in your new role, slowly ask for more responsibility or to try different areas within the project.
You will be surprised how far you get by simply asking to help in different areas.
As you learn the different parts of the job sites, you are building skills necessary to see what is needed in an entire project.
Here is the scary part. You may need to work in a few different companies to gain the proper perspective. What I mean is that every company will do things different ways and it is important to learn the good and the bad from each. It will make you that much more rounded.
While doing all of this, it is good to get a project management certificate through various colleges. There are many available at different levels. These can assist you in gaining the knowledge you need.
Then, you need the experience. This is where you will have to work your way into the project management team at some capacity. This could be a helping hand or even shadowing them for a few months.