Here is the melody, in close to its simplest form (chords shown above):
 Dm      C         Dm                 F      C         Dm   C   Dm
E-----------------|-----------------|-----------------|-----------------|
B-----------------|-----------------|-1---------------|-----------------|
G-----------------|-----------0-2---|---2-0-2-0-------|-------2-0-------|
D-0-0--0-2-0------|-0---0-2-3-------|-----------2-----|-0-2-3-----2--0--|
A------------2-3--|-----------------|-------------3---|--------------3/5|
E-----------------|-----------------|-----------------|-----------------|

So:
Dm              C
Peaches in the summertime,
Dm            Dm
Apples in the fall,
F                  C
If I can't get the girl I love,
Dm         C       Dm
Won't have none at all.

Chorus:

Dm            C
Shady grove, my true love,
Dm            Dm
Shady grove, I know,
F            C
Shady grove, my true love,
    Dm            C      Dm
I'm bound for the shady grove.


Wish I had a banjo string
Made of golden twine
Every tune I'd play on it
I wish that girl were mine

[chorus]

Wish I had a needle and thread
Fine as I could sew
I'd sew that pretty girl to my side
And down the road I'd go

[chorus]

Some come here to fiddle and dance
Some come here to tarry
Some come here to fiddle and dance
I come here to marry

[chorus]

Every night when I go home
My wife, I try to please her
The more I try, the worse she gets
Damned if I don't leave her

[chorus]

Fly around, my pretty little miss
Fly around, my Daisy
Fly around, my pretty little miss
Nearly drive me crazy

Like many old folk tunes, this song has had many verses over the years. Look em up if 
want! Hey, even make up new ones, that is how the song was created anyway!