While this method is not going to give you professional experience, it will get you started down that road. Some people take a couple of years of school just to learn this area of computers and even then it is only an introduction. However, this method should be fun.
The first thing you should do is install a Linux distribution onto an external hard drive on your computer. You should check out the possibility of problems existing with your install as some computer brands and operating systems can be troublesome. I suggest a distro such as Linux Mint or Ubuntu.
You should take some time to become familiar with Linux on the command line before installing the server software. There are lots of things you can do that will make your system vulnerable if you do not know what you are doing.
The next step, or maybe your first step is to buy at least one Raspberry Pi and start making your own projects. You will need your own ethernet cable or a USB wifi device so you can connect the RPi to the Web and your network. You can make your RPi into a server so you can learn it that way as well. Again, learning the command line and Linux in general before venturing too far along the networking path will help.
Other things you can do are to buy a second hand wireless router so you can set up your own LAN separate from your ISP provided equipment. You can connect it to your Internet or not. You can also buy an Arduino with an Ethernet shield or an ESP8266 module or an Arduino Wifi. Doing so and really learning how it works should familiarize you with the protocol used to connect it to the Web.
The Raspberry Pi and Arduino have their own websites so you can look up information on those devices there. Here are some good websites for projects:
Instructables,
Arduino,
Processing,
Hackaday,
Make magazine,
Nuts And Volts,
Raspberry Pi,
Linux,
Linux Mint, and
Ubuntu.
Here is a site that provides information on training for Linux:
http://training.linuxfoundation.org/
Some of it is free, other training costs. Regardless, much of the Raspberry Pi stuff can be found for free as tutorials and on forums.