How and what to do in order to advertise a new app? Should we start with the free version and When is the best time to set up a price for the app?

Well done for creating the app. Great question.
Regarding setting the price: I would try run a survey with some screen shots that show the game and it's purpose/experience, and ask potential users how much they would pay. I would run one survey with open end questions, such as: "which payment method would you prefer? Answers: one time payment, download payment, monthly payments, yearly payments....
and "How much would you be willing to pay for the service - in each of the various payment options".
Then, I'd run another survey with the same questions, but provide various options for them to choose from. There is a lot of psychology behind the way the questions are done, so it is important to do them correctly.
Once you've done these survey's, you'll have a better understanding of what you users are willing to pay, and how. Also, if you do the survey right, you could collect potential first clients (they would leave their email address and you can notify them when you launch).
Good luck,
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The best way to start an app advertisement is through social media it will help you reach out to a lot of people with little time and I think its best you make the app free for now but if you wanna set a price for it I know ways to make it easy for you, I also have a platform where I help individuals who are looking for ways to promote their product and services ,If you have anymore questions am always available.

Based on your description of the app I would suggest that you need some users before you can even plan for charging them. Once you have some initial users they will do the word of mouth marketing and help you spread the word faster. So take in account the money saved in marketing.

Charging or not charging also really depends how much cash your company has and how much longer your startup can survive without making any money from it. Whatever is the time estimate, deduct 3 to 6 months from it and start charging. Timing this is the essential part of the strategy.

I think its best you make the app free for now. The good idea is firstly you pay money to advertise your apps.

You can use Facebook, Chartboost ads to get your apps advertised.
Post your apps in Android forums like warrior forum, making money with Android, etc.
Post your apps on SNS networks. Making sure who is your target users, and then join this kind of group on Facebook, Linkedin, Google plus, introduce your apps to as many people as you can.
Build a website about your apps.
Choosing the right keyword on Google play store(ASO).
If you know some professionals, ask them to write some positive articles.
Asking your existing users for feedback, then encourage them to rate your apps.
Pay money to advertise your apps. You can use Facebook, Chartboost ads to get your apps advertised.
Exchange traffic(Ads exchange) with some other developers.

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