What is the easiest way to get started using databases with real data?
Charles,

You mentioned MS Excel in your comment so it's pretty much safe to assume you're in a Microsoft environment. You definitely have much power if you know how to mess with a database management system.

If you're doing some serious data analysis, I'd say go for enterprise databases like Oracle, SQL Server, MySQL, DB2, etc., which are Relational Databases. It's good to note that there are also non-relational databases out there that are gaining traction in the database market.

Since you're probably on a Microsoft environment, I suggest you go for SQL Server. You can ask your IT department if you have an enterprise database already in place. If none, you can download the "express" version of SQL Server (see link on the other comment). While you're there read about the limitation of an express edition. The express edition is FREE.

The express edition is a fully functioning, production-ready version of the SQL Server although it's limited in some ways (storage capacity, memory usage, etc). You can even run a Reporting Services off the express version. Look for "SQL Server Express with Advanced Services (contains the database engine, Express Tools, Reporting Services, and Full Text Search" --> download here: http://www.microsoft.com/sqlserver/en/us/editions/2012-editions/express.aspx

If you think that there is a necessity for an enterprise-wide database in your company (assuming your company doesn't have it yet), go ahead and buy the Developer version. The developer version contains everything that the Enterprise version of SQL Server has. If you are testing what SQL Server can do in an enterprise setting, this is the version you want to get your hands on. 

Note that you cannot use the developer version for production. It's intended for testing only. The good thing is, when your company database is ready for prime time - production - you can easily "switch" the developer version to the licensed SQL Server instance. 

As you start learning SQL Server, it is just fitting to start learning TSQL, which is SQL Server's implementation of the SQL language. 

You mentioned you want to learn running some queries, so I suggest you start with:


Data Manipulation Language (DML) - SELECT, INSERT, UPDATE, DELETE,
MERGE 
Data Definition Language (DDL) - CREATE, ALTER, DROP database
objects (Tables, Views, Constraints, Index, Stored Procedure, etc)


The SQL Server 2012 Books online is also a great starting point in learning SQL Server: http://msdn.microsoft.com/en-us/library/ms130214.aspx

The SQL Server community is also a great resource. Follow me on Twitter @MarlonRibunal. Some tips on where to find necessary SQL Server skills:


Books 
Blogs

SQL Server Central 
SQL ServerPedia (Now Toad World)
List of SQL Server Bloggers (Blog Directory)

User Groups - there is this Professional Association for SQL Server (PASS). Good resource for anything about SQL Server - events, people, etc. Explore the site to learn what the organization is all about.
Events


SQL Server User Group Meetings/Meetups - Check if there's a PASS Chapter near you - http://www.sqlpass.org/PASSChapters.aspx
SQL Saturday Event - free, one day quality conference/training for SQL Server (database administraion, development, business intelligence, etc.)

#sqlhelp hashtag on Twitter - your helpline on Twitter. I suggest you start following those people that used the #sqlhelp tag - either they were asking for help related to SQL Server or answering a question.
SQL University - collection of topic-specific blogs  
SQL Server Study Group Meetup - assemble your members and study for SQL Server Certification exams or just for enhancement of SQL Server knowledge