You can download the latest SDK including adb here. When you extract this archive, there is a readme which explains how to install the SDK. I list it here for your reference.
Welcome to the Android SDK! The Android SDK archive initially contains only the basic SDK tools. It does not contain an Android platform or any third-party libraries. In fact, it doesn't even have all the tools you need to develop an application. In order to start developing applications, you must install the Platform-tools and at least one version of the Android platform, using the SDK Manager. Platform-tools contains build tools that are periodically updated to support new features in the Android platform (which is why they are separate from basic SDK tools), including adb, dexdump, and others. To install Platform-tools, Android platforms and other add-ons, you must have an Internet connection, so if you plan to use the SDK while offline, please make sure to download the necessary components while online. To start the SDK Manager, please execute the program "android". From the command-line you can also directly trigger an update by executing: tools/android update sdk --no-ui Tip: use --help to see the various command-line options. For more information, please consult the Android web site at http://developer.android.com/sdk/
Once installed (you will have to install Java if you don’t have it already), you can navigate to android-sdk-linux/platform-tools and execute adb from there by typing