I just happen to have performed this song recently at a 150 year pioneer celebration at our (Mormon/LDS) church. I don't have an authoritative guide to the complete background but here is what I do know:

The tune is set to a Scottish dance tune called the Cumberland Reel. A colleague of mine who teaches Scottish Country Dance (to one of my daughters) supplied the name of the tune when I heard it at the class. I found the music on the Ceolas web site. Note that only the first part of the tune was used for the song. Each verse repeats the melody twice and the chorus repeats it once more.

I have a book which was published by the LDS Church Education System for the home study seminary program, which my wife got as a teenager. It contains about 15 or 20 Mormon folk songs including this one (and a record of one verse each). The book is not copyrighted and I intend to someday submit the whole thing to Mudcat/DT. Yes, I know what the road to hell is paved with.

The Handcart song, as this one is called, has six verses, appearing to be genuine 1850's words. The current version of the song sung by the children in Primary (the children's organization) was written I think in the 1950's.

I have these verses memorized, will post more after lunch:

Ye Saints who dwell on Europe's shore,
Prepare yourselves for many more
To leave behind your native lands
For sure God's judgements are at hand.
For you must cross the raging main
before the promised land you gain,
then with the faithful make a start
to cross the plains with your handcart.

Chorus:
For some must push and some must pull,
As we go marching up the hill.
For merrily on our way we go,
Until we reach the valley-O

But some will say it is too bad,
the saints upon the foot to pad,
And more than that, to pull a load
, as they go marching o'er the road.
But then we say it is the plan,
to gather up the best of men,
and women too, for none but they
Will ever travel in this way.

(Chorus)

As o'er the road the carts were pulled,
'Twould very much surprise the world
To see the old and feeble dame
Thus lend a hand to pull the same
And maidens fair will dance and sing,
Young men more happy than a king,
And children too will laugh and play,
Their strength increasing day by day.

Of the modern version, I can only remember the first verse (there might not be any more):

When pioneers went to the west,
with courage strong they faced the test
They pushed their handcarts all day long
And as they pushed they sang this song:

(chorus).

Background on handcart companies - There were ten handcart companies in all, from 1856 to 1860. Although food was always in short supply, eight of these companies came through with no more loss of life than a comparable wagon company would, and less in some instances. They were also said to be as fast or faster than any other means of crossing the plains - apparently humans can pull handcarts faster than oxen can pull wagons. The other two handcart companies, called the Martin and Willy after their captains, left late (August/September) from Iowa, and were caught in early winter storms in Wyoming, suffering much loss of life and being rescued heroically by men sent from Salt Lake (the valley referred to in the song is the Salt Lake Valley). By the 1860s, the LDS Church was financially able to send professional wagon trains to aid the converts to cross the plains and in 1869 the railroad was completed which ended the "pioneer" era as far as immigration to Utah goes.

In 1997 a 150 year anniversary reenactment of the pioneer journey was staged, where a company of wagons and a company of handcarts followed the route as closely as possible from Florence (Winter Quarters), Nebraska to Salt Lake City, Utah. They left in April and arrived in SLC July 22nd. People could come and go as they pleased, but many made the whole trip. Those who desired were allowed to march in the Pioneer Day parade on July 24th. I was at the parade with my family, and believe me, to see those folks coming up the street (they were last in the parade), both wagons and handcarts, was really something. We gave them a standing ovation which probably lasted 20 minutes or more as they passed.