Best way to administer all my passwords

The easiest way would be using several databases. Tools like KeePass allow you to store passwords in different files. You could for instance opt to store your passwords in seperate databases based on client or purpose.

For your exact use case, I personally use Lastpass. The killer features of Lastpass include the ability to compartmentalize and separate each client's passwords from each other and from my own. They also have in my opinion the best browser addon integration and the data can be stored and used offline as well.

I like to increase the password PBKDF2 iterations as that makes me just a bit harder to crack than the average user, and I absolutely will not use the service on a computer that I do not control.

Six months ago, I was in the same situation as you are right now. I used to save all passwords in my MySQL database (not safe at all) and I made a Python script to extract them. That was not a good way to solve the purpose at all.
Now I use 1password to manage and secure all my passwords in a single place and encrypt them with a master password.