Well, FTHCM is non-denominational if you were wondering.
As for AME and AMEZ. They are siblings. AMEZ is a off shoot of AME. It happened as a lot of things in this country did, because of racial inequality.
A= African. Denotes ethnicity or the roots of the members
M=Methodist. Denotes the way in which the service is organized and how orderly things are. Methodist implies a method and nearly all churches with Methodist in the name practice similar orders of service and other similarities.
E=Episcopal. Denotes the order of hierarchy. Episcopals are siblings of Catholics (not to be confused with Roman or Orthodox Catholics which are slightly different). The Episcopal order of hierarchy relates to the leadership and the titles by which they are called. Episcopals do not have a Pope. Also, it is not exactly like the Episcopal church but similar to it.
Z=Zion. Denotes the division or schizm in the AME church that caused the off shoot to add the Zion. I'm not sure why they use Zion.
The AME church is an off shoot itself from the Methodist Church.
What I just typed above is a general brief paraphrase (forgive me for any errs) of things you can read more about here:
http://en.wikipedia.org/wiki/African_Methodist_Episcopal_Church
http://en.wikipedia.org/wiki/African_Methodist_Episcopal_Zion_Church
http://en.wikipedia.org/wiki/Methodism
http://en.wikipedia.org/wiki/Episcopal
http://www.amez.org/news/amezion/aboutourchurch.html
http://www.ame-church.com/about-us/
**(Wikipedia info read it with the caveat that wiki is untested and is not a scholarly source. I added it for a quick reference. This is not where I learn my information, it is only a quick reference. Please go to a library or an online college reference library for more accurate info and references to books that have the historical information in depth.)
ETA:
AME and AMEZ both practice infant baptism. I can't think of too many denominations that don't practice infant baptism. There is a thread in the Christian Forum about baptism and infant baptism.
http://www.longhaircareforum.com/showthread.php?t=226673
http://www.longhaircareforum.com/showthread.php?t=169921