Something like Harry Potter but isn't Harry Potter.

[Bonafide Bookworm]

It could be just one single book, or a whole series.

[Heather]

Earthsea? By LeGuin?

[Johanne]

Magisterium series by Holly Black and Cassandra Clare

[Scott]

His Dark Materials

[Heather]

Magicians by Grossman

[Emily]

Here's a list that might help:
What to Read after Harry Potter

And also
Matilda
The Alchemyst "Secrets of the Immortal Nicholas Flamel" series
Inkheart series

[Katie]

Harley Merlin series

[...cats?]

Chrestomanci series by Diana Wynne Jones
Carry On by Rainbow Rowell
Children of the Red King series by Jenny Nimmo
Gemma Doyle series by Libba Bray
The Wizard of London by Mercedes Lackey
School for Good and Evil series by Soman Chainani

[Gillian]

On the magical school side of things there is:
Marked by P.C. Cast and Kristin Cast (first in the House of Night series)
Midnight for Charlie Bone by Jenny Nimmo (first in The Children of the Red King series)

For crossworld fantasy (character thrown into another world) there is:
The Twelve Kingdoms: Sea of Shadow by Fuyumi Ono (first in the incomplete The Twelve Kingdoms series)
The Divide by Elizabeth Kay (first in The Divide series)
Inkheart by Cornelia Funke (first in the Inkworld series)Tamora Pierce
Gregor the Overlander by Suzanne Collins (first in the Underland Chronicles series)

For generally good fantasy there is:
The Lightning Thief by Rick Riordan (first in the Percy Jackson and the Olympians series)
Anything by Tamora Pierce
Storm Front by Jim Butcher (first in The Dresden Files series)