Peggy, it sounds like you might be having trouble with the difference between "1st cousin, once removed" and "2nd cousin". The two are confused by people all the time.
It's really not that hard. 2nd cousins are in the same generation. While 1st cousins share a pair of grandparents, 2nd cousins share a pair of great-grandparents.
So while you have two sets of 1st cousins, one set for each parent, you have FOUR sets of 2nd cousins, one for each grandparent. A given grandparent's siblings are your great-aunts and great-uncles on that side, and THEIR grandchildren are your 2nd cousins.
1st cousin, once removed (1C1R) is more complicated, because it means TWO different things. It refers to relatives that are one generation apart, for a start. The first type is simply the children of your 1st cousins. The second type is the 1st cousins of your parents. Note that for the people who are a 1C1R of the first type to YOU, that you are a 1C1R of the second type to THEM.
The uncomplicated thing about it is that it's a mutual relation, that way. You're a 1C1R to all your 1C1Rs! That's in contrast to, for example, your aunt, to whom you are her niece - two different relationship names, depending on the perspective.