You can play Cookie Clicker directly in your browser at the link you provided: https://cookie-clicker2.com/.

This version is a free, web-based idle game that requires no download or registration. In addition to the version at that URL, here are the other primary ways to play:

Official Web Version: The original game created by Orteil is hosted at dashnet.org.

Steam: A paid version is available on Steam which includes music by C418, cloud saves, and achievements.

Mobile: There is an official Cookie Clicker app available for Android devices on the Google Play Store.

Quick Tips for Getting Started:
The Early Grind: Focus on clicking the giant cookie manually until you can afford your first few Cursors and Grandmas to automate the process.

Efficiency is Key: When deciding what to buy next, compare the cost of an item to how many "Cookies Per Second" (CPS) it adds. Sometimes saving up for a more expensive building like a Farm or Mine is more efficient than buying many cheaper ones.

Upgrades: Don't forget the "Upgrades" section in the store; these often double the effectiveness of your existing buildings or your manual clicks.

Ascension: Once you have baked a massive amount of cookies (usually in the billions or trillions), you can "Ascend" to reset your progress in exchange for Prestige levels and Heavenly Chips, which grant permanent power-ups for your next run.