Bash is a shell. It's a command-line environment, much like DOS (except written with the intent of being useful).
To use bash, you need it installed on a computer. You can install it on a Windows box, though I'm not sure you'd really want to. Nevertheless, it can be done via cygwin.
As Yitzchak has pointed out, Telnet is used to remotely access machines. It is widely accepted as insecure. It's a lot like VNC or RDP, but for command lines. If you want to access a command line remotely, you're best bet is to use an SSH service. SSH is widely accepted as reasonably secure.
To get started learning bash, you should set up a Linux box somewhere. You could do this as a virtual machine hosted on your workstation. Alternatively, you could use a Linux machine hosted by somewhere else. Many web hosting services don't provide ssh access - so you may want to watch for that.
Here's a helpful introduction to simple bash commands. This looks okay too. Oh, and this as well.
Remote access over ssh
To connect to a Linux box using SSH, Windows users can use the outstanding application, Putty. If you're on OSX, then just open the Terminal application (Applications > Utilities > Terminal) and run the command "ssh [username]@[IP-address]".
If you're connecting to an Ubuntu Linux box, you may first need to install the ssh service. From the command line on an Ubuntu box, you can install ssh service with the command "sudo apt-get install ssh".
If you want to just run a local copy of Linux, I suggest you try Ubuntu. You can install it along side Windows (just burn the CD and insert it while running Windows, the installer will guide you). Or you could install it within a Virtual Machine using Virtual Box. Once you're feeling comfortable, you could explore running a web server with it using Apache or nginx. A good project would be to set up something like Wordpress as a first website on there. You can use a VM hosting service like Amazon EC2 or Linode to run a remote Linux server for you too.
Have fun! Welcome to actual computing.