Hey, @buddawiggi! If you're looking for fossils in New Hampshire, you better look closely, because there aren't many. To find them, you would have to go deep into forested areas.

The types of fossils that have been found are from the Paleozoic Era, about 345 million years ago! If you're lucky, you might dig up some marine invertebrates, brachiopods or corals. Brachipods are by far the most interesting in my opinion. They are like snails, only with two shells. These creatures went out of existence in the Permian Period, which was late in the Paleozoic Era. I wish you luck in finding some of these rare gems!

This is a blue brachiopod fossil that was found in Kentucky:
http://3.bp.blogspot.com/_ZHv_I4h7INY/SckNYZMSr-I/AAAAAAAAAf8/4OzV0IrKjrA/s400/blue_brachiopod.jpg

NH is not a great a great place to go fossil hunting. The reason being, is that the bedrock in nearly all of the state is made up of metamorphic and igneous rocks whereas most fossils are found in sedimentary rocks. Igneous rocks are rocks that are formed by molten rock (magma) welling up from within the earth and solidifying on the surface. These include basalts, obsidian, and granites. In NH, you're likely to find many different types of granites, or granitic rock. Metamorphic rocks are rocks that have been changed do to being subjected to intense heat and pressure for a period of time. These include rock types like gneiss, marble, slate, schist, to name a few. In NH, you're likely to find various types of gneiss and schist.

As you can imagine, any rock formed form magma, or any rock subjected to a lot of heat and pressure, isn't going to contain fossils. However, some sedimentary rocks that have experienced low-grade metamorphisim may have fossils that have been slightly altered, but otherwise preserved. You might check any slate quarries that still exist in the state. For instance, there was some slate quarrying in the Mount Monadnock area. Slate is often very fine grained, so if any fossils do exist, they would likely be of good quality. Otherwise, the few fossils that can be found in NH, are in rocks and sediments of the Tertiary period (1.64 to 65 million years ago). There are plant and pollen fossils which resemble species growing in arctic Canada.