Learn Turkish: İyi akşamlar, iyi geceler.

Yes, literally they mean "good evenings" and "good nights." In Turkish, many common greetings use the plural form to make them sound more generous, abundant, and polite. You will also see this in iyi günler (good day) and tebrikler (congratulations).

Why is the plural ending -lar on akşam but -ler on gece?

This is a perfect example of Turkish vowel harmony. Akşam has thick, back vowels (a), so it takes the back-vowel suffix -lar. Gece has thin, front vowels (e), so it takes the front-vowel suffix -ler.

When exactly do I use iyi akşamlar versus iyi geceler?

Use iyi akşamlar as both a greeting ("hello") and a goodbye when it is evening time. Use iyi geceler strictly for saying goodbye late at night, or right before someone goes to sleep.

How do I pronounce the ş in akşamlar and the c in geceler?

The ş in Turkish sounds exactly like the "sh" in the English word "shoe." The c sounds like the English "j" in "jump," so gece is pronounced like "geh-jeh."