A: It sounds like you have no drip edge on your roof. Most gutters get tucked in behind the drip edge and eliminate this problem.

Assuming that you "have" a drip edge, I usually overhang about 1/2" to 5/8" past the drip edge with my starters and first course of shingles. It shows how to do it on the shingle bundle wrapper.
A: There are slight different type of drip edges http://hardware.hardwarestore.com/27-108-drip-edge.aspx.
The lower edge of the roof is the edge that first row of shingle is laying on. The drip edge should be put underneath of felt (memberance).
The rake edge (sloped edge) is two arm of the triangle. Those shingles have been nailed on the roof. The drip edge should be installed between felt and shingle.
Here is a installation instruction http://www.rollex.com/pdf/dripedge.pdf
Another quick way to fix the problem is to put vinyl gutter cover sold in Homedepot or Lowes underneath of felt. It is not necessary to nail the gutter cover unless you want to.