This is a quintessential example of American salon music from the turn of the century. Its musical distinctiveness lies in its uncomplicated diatonic harmonies and folk-influenced melodic structure.
Focus on bringing out the lyrical quality of the top voice while keeping the chords light and supportive.
Comparable difficulty and character — good candidates for what to learn next.
'Neath the Old Oak by the Mill is rated Henle level 2 of 9 — beginner repertoire, roughly ABRSM grades 2–3. In the RCM system it corresponds to about level 2.
You should be comfortable at Henle level 2 — meaning you can already play level 1–2 pieces cleanly. Learning it one level early is possible as a stretch piece, but more than that usually leads to months of frustration.
A typical performance of 'Neath the Old Oak by the Mill lasts about 3 minutes.
'Neath the Old Oak by the Mill is available as free public-domain sheet music on IMSLP — the score is linked directly from this page.
Pieces at a comparable difficulty and character include Believe Me, If All those Endearing Young Charms by Irish Folk Music, Blue Moon by Lee S. Roberts, Alice in Wonderland by Ida Hampden — see the full similar-pieces list on this page.