Please read the following first: How do I set up Android ADB?

Configuring USB access

First, you don't need root permissions to run ADB if you have set the permissions accordingly. The AOSP site provides examples for Configuring USB Access for Pixel/Nexus devices. You can use the following scheme for non-Nexus devices:

SUBSYSTEM=="usb", ATTR{idVendor}=="xxxx", ATTR{idProduct}=="xxxx", MODE="0600", OWNER="<username>"

Vendor and product ID can be obtained by using lsusb and searching for your device. (Note: The device actually is a Galaxy Nexus.)

$ lsusb | grep -i samsung
Bus 002 Device 103: ID 04e8:6860 Samsung Electronics Co., Ltd GT-I9100 Phone [Galaxy S II]

So my /etc/udev/rules.d/51-android.rules udev rule looks like this:

# adb protocol on maguro/toro (Galaxy Nexus)
SUBSYSTEM=="usb", ATTR{idVendor}=="04e8", ATTR{idProduct}=="6860", MODE="0600", OWNER="username"

You may want to repeat this for the other phone states like fastboot if needed. To let the changes take effect either reload udev (sudo udevadm control --reload) or reboot (recommended).

Making programs available

Instead of modifying the path variable I would suggest adding a symbolic link to the executable either in ~/bin/ (user only) or in /usr/local/bin/ (system wide).

mkdir ~/bin/
ln -s /opt/android-sdk-linux/platform-tools/adb ~/bin/

The directory contains a file called android, so presumably what they mean is run ./android sdk (unless . happens to part of your PATH). Also see: How to add a directory to the PATH?, so that you don't need to navigate there and run ./android.