I'm a beginner myself, but I think I can answer your questions.
Ruby gems is a ruby package manager, through which you can install various versions of Rails and other packages. A gem is a package.
Rails comes with a server. To get started, first run 'rails test', which will create a test directory, and everything inside it you'll need. Then 'cd test'. then 'ruby script/server' will start the web server (called WEBrick), by default on port 3000.
If everything works on the first try, you're very lucky. Different versions of Ruby/Gems/Rails have various issues, and you might end up doing a lot of googling to get everything square away. Also, as of some version, the default database for Rails is now SQLite 3. You can still use MySQL, but you need to pass some option to the 'rails' command, and I don't know it off hand.