First you need to know what type of server you are going to run.
Microsoft SQL Server, Oracle, IBM DB2, Oracle MySQL.
Once you make that decision, then you will need to choose the correct edition. If you are using SQL Server and you want to put it on your local machine, then you will probably just want the Express Edition.
You will then need to install it and you will be ready to use it.
From that point, you will need to know the tool that is used to interact with the server. For SQL Server it would be Management Studio (SSMS). You would use this tool to connect to the database.
After you are connected, there are many tasks that you can do by simply right-clicking in SSMS.
The next steps would be to Create a Database. Either from scratch, or some kind of sample database. http://msftdbprodsamples.codeplex.com/
Then you will need to know some SQL. SQL is the language that is used to talk to a relational database. The databases mentioned above are all examples of relational databases.
At that point you can spend your career learning more about the database.