13-15 sounds about right to me. That's enough to cover all the big concepts without overloading the PHB. My line up would be:
Human - classic, has to be there
Elf - classic, has to be there
Dwarf - classic, has to be there
Halfling - classic, has to be there
Gnome - classic, has to be there
Orc - modern classic, has to be there
Dragonborn - modern classic, has to be there
Tiefling - modern classic, has to be there
Aasimar - holy mirror of the tiefling
Warforged - built, robotic type
Changeling - shapeshifter
Fairy - small, tricky, cute, flyer
Tabaxi - Cat based furry species
Some Canine based species - Canine furry species
Aarkoccra - Bird based species, flyer