How do I set up Android ADB?

Install adb & fastboot via PPA

You can install from the WebUpd8 PPA which support both 32-bit and 64-bit. Granted you can install directly from google. But I prefer to use PPAs to keep everything up to date.
Works for Ubuntu 11.04, 11.10 and 12.04.

sudo add-apt-repository ppa:nilarimogard/webupd8
sudo apt-get update
sudo apt-get install android-tools-adb android-tools-fastboot

Source: Install ADB And Fastboot Android Tools In Ubuntu Via PPA

Install Android SDK via PPA

If you want to install the SDK which will allow you to update everything you can install via the upubuntu ppa.

So first thing is to add the ppa.

sudo add-apt-repository ppa:upubuntu-com/sdk
sudo apt-get update
sudo apt-get install android-sdk

Only issue at this point is that android-sdk needs to be run as root. Well the link in the menu didn't do much. So I updated the link in the menu, you can also run:

gksu android-sdk

From there you will need to install "Android SDK Platform-Tools" I also recommend updating "Android SDK Tools". You can deselect anything else you don't need.

Now you will need to set your environmental variables.
First you will need to make everything executable.

sudo chmod -R 755 /root/android-sdk-linux

Now to add the variables.

nano ~/.bashrc

Add these lines (at the top)

#AndroidDev PATH
export PATH=${PATH}:/root/android-sdk-linux/tools
export PATH=${PATH}:/root/android-sdk-linux/platform-tools

Source: Install Android SDK Manager (Revision 20) From PPA On Ubuntu 12.04/Linux Mint 13
Source: AndroidSDK - Community Ubuntu Documentation

Ubuntu 14.04+

Since Trusty the android-tools-adb and android-tools-fastboot packages in the Universe repository provide, respectively, adb and fastboot.

To install them:

• Enable the Universe repository: sudo add-apt-repository universe
• Update the APT cache: sudo apt-get update

To install adb:

sudo apt-get install android-tools-adb

To install fastboot:

sudo apt-get install android-tools-fastboot