“My bag just split. . . brand-new and all. . .“ - Cedric Diggory (GF20)
Students typically carry bags filled with books, parchment, quills, homework, wands, potions ingredients and ink.
Harry's bag is split in two when it is grabbed by a dwarf in Chamber of Secrets (CS13). The dwarf tried to keep Harry from escaping, insisting on delivering the singing valentine Ginny had sent him.
Ron used his bag as a sleeping place for Scabbers in Prisoner of Azkaban (PA8), whereas Hermiones bag split due to all the books she was carrying with her for all the subjects she follows that year (PA7). Harry likes to keep his invisibility cloak in it from Prisoner of Azkaban onwards (PA14, GF26), and is in fact instructed to carry it with him at all times by Dumbledore in Half-Blood Prince (HBP5).
In Goblet of Fire, Harry split Cedrics brand new bag (GF20), to inform him about the dragons for the first task. Harry used his own bag to carry chicken legs, bread and juice for Sirius to Hogsmeade (GF27).
Bags were also very useful to protect against attacks from Peeves, as they were used by students to cover their head (OP18).
Luna liked to keep odd things in her bag, such as a Gurdyroot, toadstool and cat litter (HBP20).
Several times in the books, Rowling refers to "bags and books" (e.g. GF15), suggesting that the books aren't kept inside the bags. Even though at other times, she clearly refers to Hermione, Ron or Harry stuffing their books inside their bags (e.g. GF14).
In Goblet of Fire the word schoolbag is used for the first time (in GF 18 as school bag, and from GF37 as schoolbag). So it seems like there is no standard bag for students going to Hogwarts. It is alsno never mentioned in the books where Harry got his bag in the first place, and how he got another one when his bag was ripped in Chamber of Secrets. Also, most students swing their bag on to one shoulder (OP25), which means it is not a backpack.