This table will tell you which birthstones go with which months:

Birth Month
Modern Birthstones
Traditional Birthstones
Mystical Birthstones
Ayurvedic Birthstones
January
Garnet
Garnet
Emerald
Garnet
February
Amethyst
Amethyst
Bloodstone
Amethyst
March
Aquamarine
Bloodstone
Jade
Bloodstone
April
Diamond
Diamond
Opal
Diamond
May
Emerald
Emerald
Sapphire
Agate
June
Pearl, Moonstone
Alexandrite
Moonstone
Pearl
July
Ruby
Ruby
Ruby
Ruby
August
Peridot
Sardonyx
Diamond
Sapphire
September
Sapphire
Sapphire
Agate
Moonstone
October
Opal, Tourmaline
Tourmaline
Jasper
Opal
November
Yellow Topaz, Citrine
Citrine
Pearl
Topaz
December
Blue Topaz, Turquoise
Zircon, Turquoise, Lapis Lazuli
Onyx
Ruby

The Modern Birthstone list contains the official birthstone list from the American National Association of Jewelers, Jewelers of America. These gemstones were officially adopted in 1912. In the US, this is the accepted list.
The Traditional Birthstone list contains older birthstones and is sometimes combined with the modern birthstone list. These stones reflect societal birthstone traditions, heralding back to the 15th century. (Many jewelers have differing lists of traditional birthstones).

The Mystical Birthstone list is of Tibetan origin from over a thousand years ago.

The Ayurvedic Birthstone list is from Ayurvedic Indian Medicine – a type of medicine used for over a thousand years.