how to pronounce "t's" sound, as in "it's" or "that's"

Can you pronounce the T and S sounds independently? Then, can you pronounce "TSSSSS"? T's isn't a special writing combination in English the way CH is, it's just those T and S sounds in sequence.

Technically, you would touch your tongue not to your palate but to the alveolar ridge, the bit of hard meaty tissue that's just behind your upper front teeth.

If by Z your friend happened to mean "the way the letter Z is pronounced in German or Pinyin", that's pretty much spot on. If she meant the way it's pronounced in most languages, then no.

The pronunciation is different from a common Z sound in two ways: you need to briefly stop the passage of air through your mouth in order to make the T part (in that way it's more like a DZ than a Z) and you do not vibrate your vocal cords (in that way it's more like a SS, or a "whispered" Z than like a normal Z).