How do I find the Oracle database version?

You can check the Oracle version by running a query from the command prompt. The version information is stored in a table called v$version.

How do I check my database client version?

In Windows. Check the Inst_loc entry value which will be the software installed location. You can use command prompt or you can navigate/explore to the oracle home location and then cd to bin directory to lauch sqlplus which will give you the client version information.