I think that Shape Water can both target and affect liquid water and solid water, including snow, but we’ll get to that later.
The targeting area is a 5’ cube, but you can animate that into any shape you want so long as it doesn’t travel more than 5’ per casting. This animation lasts one hour, so you can shape it (like in the name) all you want in the meantime, including outside of the 5’ cube you initially selected because:
• you can animate the selected volume of water that fits into a 5’ targeting cube
• you can animate that into a sphere
• water is incompressible, so the sphere would have the same volume as your water in the target cube
• because this volume of water is animated into whatever shape you want for an hour and does not require your concentration, it does not have to fight gravity and can hold its own shape
• the spell also states clearly that you can change the shape to animate at your direction. As you have already cast this spell, and the animation lasts for an hour, the verbal and somatic components are done with the casting and now unnecessary, so we can agree this is mental and all else is flavor
• you can, as an instantaneous effect of casting this cantrip, move this volume of water up to 5’
• it is not stated that this spell teleports the water, so we can reasonably assume that the water moves continuously up to 5’ IN ANY DIRECTION (sorry, I’m on my phone and don’t know how to bold on this version of the website)
• because you can direct the water to hold its shape, you can move a 5’ cube of water within the shape even if it is the full 125ft^3 and move the entire volume because the animation and it following your direction is a persistent effect that does not require verbal or somatic components past the casting that initiated it, so this direction function that does not require concentration can not interfere with the casting to move the volume as a whole.
I believe you can target ice and snow with this spell because one of the listed effects is that you can freeze water that fits within the 5’ targeting cube, and this water remains frozen for an hour. It states plainly that the frozen water, or ice, is still water. Snow, sleet, hail, etc are all just ice in different sizes and crystalline structures, and are therefore also water by this spell’s own definitions. The spell also specifies in the targeting section that you can choose an entire “area” (volume) of a 5’ cube when casting this, meaning it doesn’t all have to be one discrete volume of water like only water in one glass at the table, so this could naturally apply to every snowflake or sleet crystal within a specified shape or volume you choose that fits within the targeting cube’s limit of 5’ because it doesn’t even have to be a cube.
Given the targeting language, I’d argue you could select a whole 5’ cube of snow, animate it into a cube for an hour, recast to move that cube or whatever simple 125ft^3 shape you made it into an entire 5’ up, and even direct it to change shapes and animate while you’re moving the volume, even while recasting to move the volume up to 5’ in any direction.
It’s basically water bending from ATLA except you have a maximum target of a 5’ cube, you can’t move selected volumes of water more than 5’ per casting, you can’t melt ice you didn’t make with this spell, but you can change its color and opacity. (Blood and swamp bending are different, levelled spells, unless you count Druidcraft.)
TL;RD — Yes, you can select up to 125ft^3 of water (including ice, snow, sleet, hail, fog, dew, steam, etc.), so long as it fits within a 5’ targeting cube, animate the volume targeted past the bounds of the targeting cube as part of your animation directions that last the entire hour, and move the volume animated up to 5’ per round in any direction as a separate casting at any point during the animation’s duration. You could reasonably use this spell to pick up a 5’ cube of water from a fountain, animate it into a simple shape, move it around at a speed of up to 5’ per casting, freeze up to a 5’ cube of it, and, if you have Aspect of the Moon or a friend with this spell, too, who can take over while you sleep, trek it hovering through the blazing hot air or along the burning sands across a desert continent with breaks to unfreeze it at will and refreeze it since you can’t freeze ice and the ice produced can only melt all at once either at the end of the hour or when you want.
*caveat: if your DM wants to be strict, you have to direct the animation back into a 5’ cube to be targeted for a recasting to prolong your animation over this water if you intend to not lose any that would be outside of the targeting cube or have it splash all over the floor at the end of the hour.