Do you have any control of the server side, or is this a generic email client? If you can control the server side, I would do something like authenticate, then have the server generate a UUID and keep that locally to future api calls. Another idea would be to send a hash of the password to api calls instead of the actual password, then you can store just the password hash locally.
The issue with encrypting the username/password is that your code needs to be able to decrypt it, and if your code can decrypt it, somebody can reverse engineer your code and do that as well, although you can make it easier/harder by how you code and package it.
Once you figure out WHAT you're storing, you can figure out how you store it. One account? Shared prefs. Multiple accounts? Create a Sqlite DB.
I would suggest using http://ormlite.com/ to handle your db connections. I did a good chunk of the initial Android port work, and its now been enhanced/maintained by a top notch group of hackers. Very solid stuff.
See more Sqlite blog posts:
http://www.touchlab.co/blog/single-sqlite-connection/
http://www.touchlab.co/blog/android-sqlite-locking/