Hi
The cost per month will depend on the location, size and quality of the apartment. For example, for a 2 bedroom apartment located in the center, you can expect to pay 600 euros per month if it is in poor condition and 1000 euros per month if it is recently renovated. A similar size apartment located outside of the center can cost about half of that.
When you sign a lease, expect to pay 1 month rent in advance plus 2 months rent for a security deposit. So, if you rent an apartment for 1000 euros per month, on the day of signing you will pay 3000 euros.
You can discuss the issue of the rental agreement with the landlord. You will find that when you apply for the blue card application, you will need to have a signed lease and the landlord will have to give "permission" for you to stay at the flat. They will do this by either coming with you to the immigration department or signing a document in advance that will need to get notarized.
Use this website:
To find apartments for rent. You can search in English and then if you see an apartment you like, you can contact the person responsible via the website. But be warned, about half of the time, the real estate agent will not return emails. If you find a flat you really like and they do not respond to the email, you have to call them.