Quick answer
Some are spelled out letter by letter (МФЦ = “em-ef-tse”), others are read as words (ЗАГС = “zaks”). There is no rule — you learn each one.
Offices and services
Transport and money
Everyday
The three to memorise first
МФЦ is where almost all paperwork happens now — one counter for registrations, documents and certificates. ЗАГС handles births, marriages and deaths. ГИБДД is the traffic police. If you live in Russia you will deal with all three.
Do they decline?
Those read as words do: в ЗАГСе, из ЗАГСа. Those spelled letter by letter normally do not: в МФЦ, из МФЦ. Gender follows the main noun inside: МГУ is masculine because университе́т is.
Why so many?
Soviet administrative language loved compound names, and the abbreviations outlived the institutions — ГАИ was renamed ГИБДД in 1998 and people still say the old one. Expect to meet both names for the same thing.
Typical mistakes
Reading ЗАГС letter by letter
“zaks”, as a word
Some acronyms are pronounced as words.
в МФЦе
в МФЦ
Letter-by-letter acronyms do not decline.
Check yourself
How is ЗАГС pronounced?