Our family, one son at 2, sing this every night. It has become a blessing to us as it is Psalm 1 and a basic introductory Psalm to all Psalms and for all of life.
Learn to sing the Psalms Brad. You will be teaching yourself, your family, your children God's Word and singing God's Word at the same time. Which means you are teaching them doctrine from God's Word and not from VBS fluff hymns/songs...
We began singing Psalms after being convicted that we should sing them in worship. I have learned to love the Psalms much more than the hymns now because I know without any doubt they are God's Word. Why sing anything but God's Word...?
Andrew and everybody are right. When I was growing up, the Anglican church in its old-fashioned way still had very little truck with "children's hymns". Children sang the great psalms and hymns of the church along with the rest of the congregation, and that being the optimum time of life for rote-learning, without even trying they would memorise and internalise the words.
Often those psalms and hymns were only dimly understood at the time, but for sensitive children they resonated even so - and more importantly they were stored away as a treasury ready to be called on when understanding dawned.
It's the same with the music, or even more so.
I mostly sang easy-to-understand hymns that were theologically rich and well-known. But if you're a fan of catechisms, and want to incorporate singing, you should check out Songs for Saplings. Fair warning (because I see you're Baptist): They're Presbyterians and use the Westminster Catechism.
We have morning and evening prayer in our home with the singing of hymns in both. We are always learning a new hymn each month. We've maintained this practice for years. It has paid off.
Among many others, my Boys know these songs well:
Doxology
Gloria Patri
Threefold Amen
Hear Our Prayer O Lord
23rd Psalm (Scottish Psalter)
O Worship the King
Come Thou Almighty King
It is Well with My Soul
Amazing Grace
Holy, Holy, Holy
This is My Father's World
Come Thou Fount
The key is consistency and repetition over many years as a family.
Adding (from the red Trinity Hymnal):
Lift High the Cross
It is Well with my Soul
A Mighty Fortress is our God
O Worship the King
All Hail the Power of Jesus' Name
There is a Fount Filled with Blood
Be Still My Soul
Be Thou My Vision
Crown Him with Many Crowns
(I guess I could go on and on!)
I have sung these intermittently with my children since they were 2-3 years old. They sing where they can.