Telephones are not connected to your mains and are safe to work on without risk of shock.

Your phone supplier will have supplied you with a master incoming box when they installed the phone. That is their property and you shouldn't attempt to move this. It is however fairly straightforward to add extension boxes to this master by simply attaching 4 core telephone cable to the master and connecting that to a new extension box.
Make sure you buy an extension not a master. They look slightly different. The master has several electronic components on it whereas the extension has none. You can connect an extension from the master or from another extension.

You can also choose between a surface mounted box or recessed. If you choose a recessed you have to buy a mounting box and dig out a hole in your wall to recess it into. Obviously the surface mounted is easier it simply screws straight onto the wall.

Choose the position you want the extension and screw it to the wall.
Run some 4 core telephone cable from the master to the extension.

There are several choices of how to deal with the cable:

Method 1 Use hammer in cable clips to attach it to the wall or skirting. Try to get the cable nice and tight and as neat as possible. You may need to drill holes through walls or doors frames where necessary.

Method 2 Buy some self adhesive mini trunking and run the cable in that.
You can cut this to size very easily with a junior hacksaw, then attach it just above the skirting. It comes in two sections. The first bit attaches to the wall. The top piece clips into place when you've got the cable inside.

Method 3 Chisel out a channel in your wall.
Mark the position of the socket and a channel for the cable. Use a cold chisel and Club hammer(heavy hammer) to chisel the channel from into the wall(quite messy).
Insert the cable. If necessary hammer in a few cable clips to hold it. Then make good the wall with filler or plaster. When dry paint over.

Screw type

Simply unscrew the terminal screws with a small screw driver and insert each cable having stripped off 5 -6 mm. of the individual insulation. Then screw them tight.

IDC (insulation displacement) type. These are quite convenient to use as long as you ensure a good connection. You don't strip any insulation from the individual cable. You can buy a special IDC insulation tool which forces the wire over the contacts on each terminal. (for a round £7).

You can also get very successful results without a special tool if you place the cable between the contacts and press it firmly in place with the sharp bit of a stanley knife blade.
For best results don't have more than 2 cables on an IDC terminal (i.e. one extension.) If you want another extension add it to the extension not the master.

The number of extensions you can have is not limited but you can only have a Maximum of 4 phones plugged in at any one time.