The nursery rhyme begins this way: Mary had a little lamb . . . And everywhere that Mary went, the lamb was sure to go? Is the correct way

Yes, Mary had a little lamb is the correct way this rhyme.

Below the correct lyrics of Mary had a little lamb nursery rhyme song.

Mary had a little lamb
Little lamb, little lamb
Mary had a little lamb
Its fleece was white as snow
And everywhere that Mary went
Mary went, Mary went
Everywhere that Mary went
The lamb was sure to go
He followed her to school one day
School one day, school one day
He followed her to school one day
Which was against the rule
It made the children laugh and play
Laugh and play, laugh and play
It made the children laugh and play
To see a lamb at school
And so the teacher turned him out
Turned him out, turned him out
And so the teacher turned him out
But still he lingered near
And waited patiently
Patiently, patiently
And wai-aited patiently
Til Mary did appear
Mary had a little lamb
Little lamb, little lamb
Mary had a little lamb
Its fleece was white as snow
And everywhere that Mary went
Mary went, Mary went
Everywhere that Mary went
The lamb was sure to go