This tutorial is going to get you up and running with Unity 5, focusing on how to download and install Unity 5.
I'm only able to give instructions for Windows/PC, so you won't find details on how to do this with a Mac, but once you see how simple this is on Windows, it shouldn't be too difficult to apply the same process.
Before we get started, do you know the difference between Unity 5's Personal Edition and Pro Editions? Really short version: Personal is free; Pro has a fee! You can find out more information at the download link below:
You no longer download the full installer straight from the website, you download a small program which will manage the download and installation for you, that's the first step.
So now you have downloaded the Unity Editor (see: How to get the Unity Editor), it's time to install it. As with the downloading, installing Unity is a pretty straight forward process. Follow these steps to install Unity:
While it's downloading, you can move on to creating a Unity Account...
In this section, we'll go over creating a new Unity Account. Now, as you'll notice when you first launch Unity, you don't actually need an account to use Unity. There is an option to launch Unity as a guest, and not provide any account details. So why would you need to create an account, if you're going to use the personal version? Answer: The Unity Store. If you want to download anything from the Unity Store, you'll need to sign in with a Unity Account; personal or pro. There are free assets on the Unity Store, such as the standard assets packs, which is useful to get you up and running with basic things in your early projects. You'll also need a Unity Account to make use of the Unity Community forums, too. If you're still not convinced you need an account, then please feel free to skip on to the next section. Below are the steps needed to create a new Unity Account.
Your account is now created, confirmed and ready to be used with the Unity Editor, Unity Store and the Unity Community.
Once the installation is complete, you will see the Completing the Unity Setup screen; click Finish (Optional: Untick the "Launch Unity" tickbox first).
Unity should now be installed. If you encounter any problems with the installation, please check Unity's Community for assistance.