Before you begin playing Eve, you'll need to do the following things:
In late 2016 EVE Online switched to the Free-to-Play-Model. Here you can read about the differences between a Free-to-Play Player (called an "Alpha-Clone") and a paying Subscriber (an "Omega-Clone"). You can get your Free Account and help Goonfleet at the same time by using this link to sign up: https://secure.eveonline.com/ft/FreeTrialSignup.aspx?aid=112810
Even if you are ready to subscribe an account right away, you should consider staying Free-to-Play for some time to learn the game and play the tutorial. Some basic skills you want to learn are available as an Alpha-Clone. On the other hand, Omega-Clones learn skills faster. Your choice.
Before you subscribe, take a look on the EVE Online Website and watch for special deals for first time subscribers. In addition to your gametime, you usually can get some extra stuff like skillpoints, ships, skins etc.
The downloads for both Windows and macOS users can be found at https://community.eveonline.com/support/download/
As of March 2015 and the Scylla release, the EVE client uses a download on demand system to speed installation and get you into the game faster. This means a significantly smaller initial download, but models and art assets will be loaded in the background during gameplay, adding lag. This doesn't present a major problem for the tutorial, but will put you at a disadvantage for PvP.
You can instruct the launcher to download all the assets from the options of the launcher. It's recommended that you do this before participating in fleets. The option can be temporarily disabled to get into game faster after a patch (to change skills, update market orders, etc).
Placing the Shared Resource Cache - the location that the launcher downloads the assets to - on a solid state drive can speed up to process of loading assets, resulting in less lag.
A beta launcher was released in October 2015, allowing users to launch multiple accounts from the same launcher. The beta launcher also unifies the resource cache location across the main and test servers, meaning that users of it do not need to have more than 1 EVE client installed.
The EVE Launcher checks for a patch when first opened and will automatically update the EVE client. If the launcher has been left open over downtime, it will need to be reopened to check for patches. If you start the client without the launcher and it is out of date it will give an error about "Incompatible Build" and exit.
The EVE repair tool can (sometimes) be used to update if the launcher fails to do so, but CCP has been making major architectural changes throughout 2015, and has issued warnings on several occasions not to use the repair tool to apply certain updates. As such, the launcher is now the primary, supported way to update, and you should check patch notes and dev blogs carefully before using the repair tool to update.