Before you get started, make sure you buy Uru - Complete Chronicles (referred to as "offline Uru" below) and create an account for the Guild of Writers forums. If you are having trouble registering, feel free to contact me (see the signature below this post). Do not try to use a Myst Online: Uru Live client as basis. The old offline Uru and Myst Online: Uru Live are related but independent and different games!
You should not use an existing offline Uru installation if you have one, since it might have been changed in a way that breaks playing on a Shard. Instead, follow the installation guide. Remember that you will need to install both "Uru: Ages Beyond Myst" and the excansion pack "The Path of the Shell". Now download the Deep Island patcher. Extract the file into some folder (preferably on your Desktop) and copy its content into the directory with the clean offline Uru you just created. If you are using Windows 7, right-click onto "Uru.exe", choose the "Compatibility" tab, and select the compatibility mode for "Windows 7" (I know that this step sounds crazy, but it helped for many many people, so please just do it).
Double-click on Uru.exe and you should be asked for username and password: Use the one from the Guild of Writers forum, and hit "Login" (this is case-sensitive, use the name exactly like it is displayed in the forum). The avatar creation window will appear if you do not yet have an avatar. Please ignore the invites settings as well as the option to import an offline savegame, that does not work! Hit "OK" and then "Play" to proceed to the actual game. On your first login, an update of more than 200MiB will be downloaded, so it might take a while. If it's all working, you will end up in the avatar creation screen, and the avatar will be sent to a desert in New Mexico. Have fun!
Do not use Drizzle on this installation. Most of Drizzle is suitable for offline installations only. The only exception is the installation of ages from other Cyan games.
In case you are using Windows 10, Annabelle has written up an excellent installation guide - that should help you if you are still confused.
For Linux users, a recent version of wine should work very well to run Uru. On macOS, PlayOnMac is reported to work very well, whereas wineskin has serious problems and should be avoided.