This is a 19th Century Irish Folk song. It is a type of jig, with a fast tempo and bouncy rhythm. The Dubliners play the most famous version, mainly using a banjo. We're transcribing it to guitar here, and calming it down to be easier to learn. Once your fingers know the basic melody, embellish it all you want and put your own stamp on it. That's what Folk music is all about.
The notes are from the C major scale at open position (seen here).
Now, if you are familiar with the C major scale, the temptation is to play everything in 1 position (with the first finger always covering notes at fret 1, 2nd finger at fret 2, 3rd finger at fret 3). Because that is how the scale is usually practiced. However, in this song it is easier to play most of the song with your first finger at fret 2. Only revert to the 'normal' position during the high parts. This is because you only need 2 fingers for most of the tune, and at full speed it is somewhat easier to play between the 1st + 2nd fingers and open notes. That is the method we show here.
The picking is really important. Listen to the normal speed audio a few times until you 'get' the rhythm / tune in your head. Like most Irish music it isn't standard 4/4 time (we used 6/8 here). Listening will help you immensely when it comes down to how you pick the notes. You must have the sense of timing in your head first. The picking directions here work for me. However, you may find it more natural to use different pickstrokes for some sections of the music. Do it your way as long as it helps you get the right speed and feel.