SQLite3 is used in the Rails project to create the application's database. Its code is public domain, and is available for use privately and commercially. Our courses don't cover a lot of what you can do with SQLite, but there are a lot of great resources that go over the queries and reports you can create.
In general, the principles of database software are very similar between tools (MSQL vs SQLite vs Oracle Database etc), and the commands needed to interact with a database will be similar across each one. This means that while there aren't many SQLite3-specific tutorials out there, there are other resources you can use.
If you are looking for a program that will let you visually navigate through a database (kind of like how you would do in your computer's file system), I highly recommend Navicat. There are free and paid versions, and you can navigate through your database manually, or use SQL syntax to get the records you need.
Essential SQL - Kris Wenzel
This is a thorough tutorial on the database operations you can do on SQLite3, that assumes you do not have any background knowledge.
Learn SQL the Hard Way - Zed A. Shaw
Part of a great series on how to learn programming without a programming background. The tutorial is still in Alpha, and is only about half finished (15 lessons available out of ~30). Highly recommended, very honest about software limitations and the importance of various attributes.
Official SQLite3 Documentation - SQLite.org
Thorough, but not very user-friendly.
Basic MySQL Commands - The West Chester University Computer Science Department
Looks complicated, but this is a great reference guide to all MYSQL commands. Some of the more advanced ones may not work in SQLite3, but the vast majority will work fine.
Exploits of a Mom - Randall Munroe, XKCD
A little database humor before you go :)