How to install a new TV outlet on the opposite wall?
Somebody knows, I am sure

It's not exactly fun. A lot depends on if/where there is access. Attics and basements make it easy.

When you have neither, this can be a problem and will involve drilling some exploratory holes and patching. It might involve a boroscope. Any doorways?

So, attics? Basements?
Must be behind wall? Doorways?
Metal studs? Wooded studs?
Picture?

You can drill holes with a flexible drill bits considerable lengths.

It is an attic, but after I will cut the existing TV wire what I will do?

It usually makes a lot of sense to make all TV wires originate from one point in the house. It would be great if you can do that. That's the right way.

The low voltage plates/TV plates when you remove them, there will be a connector behind the plate attached to the plate. This F-F adapter is available separately. I usually use a right angle fitting at the outlet plate as well.

So, you could disconnect at the back of the TV outlet, use a F-F adapter and then extend the wire where it needs to be.

The "F" connectors come in 3 styles: Crimp, twist on and compression and are available for RG-59 and RG-6. Wiring between walls is always done usig RG-6. Crimp and compression require special tools. Compression connectors are preferred. The crimper is expensive and the connectors are too.

Thank you KISS, I live in KISS-immee, pale coincidence... My TV is HDTV I need a special cable?

Cool.

The cable you should really use is RG-6 QS or Quad Shield. Connectors are different for RG-6 and RG-6 QS.

This company: RG6 Quad Compression F-Type Connector with O-Ring - 20pk - Cables To Go has a 10 pk of the connectors and the crimping tool. It adds up fast. Home depot carries the connectors, crimping tool and probably the cable by the foot.

Here is basic instructions for installing the cable: F-Connectors - How to Install F-Connectors - Coaxial Cable Connectors However, this one uses a crimp and not compression, but still the instructions are similar.

Here is F-connectors - RG6, RG11 & More | ShowMeCables a list of some of the connectors and adapters I was talking about. The right angle and the F-F are there. No compression connectors.

Radio shack should have the right angle, wallplates and F-F couplers.