There was less of a distinction between children's and adults' dress than you might think (there was no age where fashion suddenly kicked in), but Victorian children did have their own particular ways of dressing. I'm going to write specifically about the 1880s, as I think that's the time period you're asking about, so what I write below isn't necessarily true of the entire Victorian period.

Babies had the most basic clothes, regardless of gender. A baby would typically be dressed in a white chemise and a white gown - white cotton or linen was the easiest thing to wash, as it could simply be boiled hard. The only thing that mattered for the fit of the gown was the baby's arms, since the length would usually be somewhat past its feet. As said baby started to walk, its dresses would be made with a skirt that would stand clear of the floor. Oh - and a white cap was also a must during infancy.

Once a baby moved into childhood, their clothing became a little more complex. On top of their chemise, they would wear some kind of supportive underbodice that would encourage good posture and, theoretically, self-control. On top of that, they would have a petticoat with a bodice (since they lacked a defined waist to fix a skirt to), and then a dress - boys and girls. This petticoat and dress would generally be about knee-length, with stockings and boots below. The lines would follow contemporary fashion in only the slightest way: as the bustle came back into style around 1883, there might be a bit of fullness put in the back of the skirt, and through the decade a dropped waistline was also very common. (There is more discussion about how this genderlessness changed in a previous answer I wrote over here.)

At about the age of three, children's clothes would start to be differentiated by gender. Boys would wear shorts or knickers with a "blouse"/tunic or a jacket-and-waistcoat in one, while girls would wear somewhat more complex dresses than before. Sailor suits were quite popular for children, with a skirt for girls and pants for boys. For both boys and girls, the amount of leg shown was an indicator of youth: when a boy went into long pants or a girl started to wear a skirt that came to the top of her foot, they were considered grown up and able to take part in courtship. With boys, this was more of a binary (short or long trousers), but for girls there was a whole spectrum of skirt lengths from the knee to the foot, as well as more and more shapely corsets. It was also common for young women's bodices to button down the back, unlike adults', and to be made with a single dart on either side of the front rather than a pair.

As far as class differences go ... it's really the same as for adults. A working-class child would probably have hand-me-downs that might be more behind the times and/or made in cheaper fabrics and with less ornamentation, while a wealthy child could be dressed very fashionably. Frances Hodgson Burnett's Sara Crewe, or What Happened at Miss Minchin's (1888), the novella that would become A Little Princess, describes the title character's wardrobe as "so grand and rich that only a very young and inexperienced man would have bought them for a mite of a child who was to be brought up in a boarding school. ... Her dresses were silk and velvet and India cashmere, her hats and bonnets were covered with bows and plumes, her small undergarments were adorned with real lace ..."