You are correct in terms of what ways are there for you to practice. However, more than how you can prepare yourself it is good to have in mind what skills you need to master and how each of the steps you listed will help you in that task.
n my perspective there are four key skills:
a) Clarifying the problem at hand
b) Structuring an approach to the problem (a.k.a. find / apply a framework to the problem)
c) Based on the approach structured, conduct the analysis
d) Draw, synthesize and communicate conclusiongs / insights / recommendation or next steps
Referring back to your first question - "What should I look to gain from each of these steps?" and the three steps you listed. Steps 1 & 2 will give you a good perspective on what is expected from a top-notch candidate in each of the four skills. Step 1 will help you with understanding what is a good and structured approach to the different problems. Step 2 will help you understand the best ways to clarify the problem at hand, how to go about your approach conducting the analyses and how to synthesize your findings, conclusions and recommendations. Finally, Step 3 will help you become fluent in all the four skills.
In terms of "When you should move to the next step?", there is no general rule, as those three steps will probably overlap (i.e. you'll probably refer back to the frameworks while practicing actual cases). Anyhow, my advice would be:
a) read the materials that you find relevant (I personally recommend Case Interview Secrets by Victor Cheng, that will cover steps 1 & 2);
b) Start practicing after having read the materials - this should take ~80% of your prep time (e.g. if you have ten days to practice, you should dedicate ~8 days for practicing cases)
I hope it helps!