When initially entering the game, there will be the familiar server list of all the possible servers you can join. These are categorized on the left. There are 3 public server categories: Official, Community, and Modded. Click on the Modded tab to show a list of available modded servers that you can join. Likely, the names of the modded server will have info on what mods they use.
No, you do not need to install any special programs or such in order to play on modded servers. The regular Rust application already has everything you need to run a modded game.(Keep in mind these are the same people who made Gmod, so modding is nothing new to the devteam)
Say I want to teleport to my friend Blaberous. I would have to use a special command targeted at my friend to request a teleport, which is in most servers the /tpr command.
After /tpr I would put down the name of the person I want to send the request to. So in this case, /tpr Blaberous. (You may have trouble tpr-ing to a person with spaces in their name.) After I sent my tp request, Blaberous will receive a message saying that I want to teleport to him. He then has 30 seconds (more or less depending on the server) to accept my tp request with the /tpa command. If he does, then I will receive a message saying he accepted my request, and in about 5-15 seconds, I will be teleported to his location. You will need to wait a brief moment of time (depends on the server) after the teleport is accepted before you actually teleport. The chat will say to you and to no one else, “Teleporting to (player name) in (X) seconds!” This next part is important. In this brief amount of time, usually about 10 seconds, avoid taking any kind of damage. Otherwise your teleport will be interrupted. The reason why the mod is made that way is so players can’t get into a fight and teleport out in the middle of it.
Again, keep in mind that these time figures are approximate, and vary by server. Also keep in mind this is for if I want to teleport to him, not vice versa.
NOTE: If your teleport target is on a ceiling, or on a rock high off the ground, he will need to relocate. This is because you can only accept teleport requests on ground level, or on a foundation.
If you want your friend to teleport to you, he will have to send a teleport request to you.