How many counts make up a typical group? (What do we most often count to when making music?)

By far, the most common way of counting music is to count four quarter-notes at a time.  No matter how long the actual notes of the song are, you should be able to “feel” (and count) four quarter-notes worth, over and over again, throughout the whole song — usually.  You’ll be saying, “1, 2, 3, 4, 1, 2, 3, 4 . . . “ over and over again.  How fast?  That depends on the tempo.  For a faster song, you’ll be saying, “1, 2, 3, 4 . . . “ fast; for a slower song you’ll be slower.