If you forgot the password of your Instagram account, it is easy to recover using your email, mobile number or facebook account. But you can't know your current password in any way. To recover password you have to do following things:
1. Open Instagram app and click on "Get help signing in".
2. Enter your Instagram user ID.
3. Click on "Send email"
4. Check your email account for email from Instagram.
5. Now, click on the login button in email received from Instagram.
6. Enter a new password.
7. Click on "Reset password"
To manage your passwords you can save them in a safe place e.g. a database in your computer or in diary.
You can also use some apps like Last pass, 1 Password, keeper, Dashlane, EnPass, Master Password etc. To manage your passwords. These apps helps to store all password at a place and online shopping profile.
But I think there may be some risk in saving all password in an App. So it's better to use a database in your computer or in a diary.