Most frequent flier programs place Zambia into the Africa zone. You can find some generic recommendations on how to get to East Africa here. You can also fly to South Africa and then get a flight from JNB to LUN. I already have a blog with my recommendations for which programs to join if you are new to the world of miles and points.
The major gateway city for Zambia is Lusaka (LUN). There are also international flights to Livingstone (LVI). Once you have been in the miles and points game for awhile, you will get a feel for which airline to use where but if you are just starting out Wikipedia will show you all the airlines that fly into Lusaka and Livingstone.
LUSAKA: No One World airlines fly here.
LIVINGSTONE: Comair (British Airways subsidiary) flies from JNB.
LUSAKA: Ethiopian flies from Addis Ababa and South African flies from JNB.
LIVINGSTONE: Ethiopian flies from Addis Ababa and South African flies from JNB.
LUSAKA & LIVINGSTONE: Kenya Airways serves both airports from Nairobi.
Emirates serves LUSAKA from their hub in Dubai with connections world wide.
HOW MANY MILES WILL IT COST?
Every airline member of the 3 main alliances has it’s own frequent flier program. Examples are in my East Africa post.