Here's the best way..
Navigate to where you've installed the sdk. Once you've gotten into the sdk, you're going to look for "PLATFORM TOOLS" folder, inside hold shift + right click on any empty space, select "OPEN COMMAND WINDOW HERE"
Connect your phone to your PC via usb cable, in the command window type "adb devices" without the quotes. If you've set everything up properly it should return an alpha-numeric value, this is your device ID. If not, you've missed a step or installed sdk in the wrong manner. Go back and double check your steps.
Now that we know you're connected there are a few options, you can run the logcat in command window, or you can export it to a .txt file.
To see the log in command window just type "adb logcat" (again no quotes) and the log will scroll through your screen. To stop scrolling so you can investigate hit ctrl + c
To export to a .txt file, type "adb logcat > logcat.txt".