couldn't grok much from the video but I'll give ya the basics ....
1. wood prep: 80/100/120 is good for removal of old surface materials, once finished wipe all pieces with either water or denatured alcohol and let dry (longer if you use water)
this will clean wood and also raise the grains.
Once dry, resand using 180/220 to remove scratches and marks left by rougher paper
wipe again with either water or denatured alcohol and let dry.
With furniture, never have sharp edges/corners, especially with children, so either do a bevel or a round over (1/8" or 1/4" would be enough)
finish: you can stain, or not , depends your preference......
if using a solvent base stain (alcohol/petroleum) you need to let pieces dry and air out for 36-48 hours to get rid of solvent gases, else the gases will blister your varnish/finish
and the gases would also be bad for newborn/children...
There are food safe stains and oils out there, used on salad bowls, etc)
I would strongly suggest looking for those brands/products at local woodworking/hobby shop.
Most food safe oil finishes last longer than varnish/shellac/lacquer, more durable, and easier to reapply as time demands.
Also, they will give the wood a richer colour, negating the need for stains unless you're trying to match something else.
Whatever you use, you'll need at least 2-3 coats for a proper seal and finish of the wood.
hope that helps.