Fossils are not usually found in igneous and metamorphic rocks that have been very hot, like granite and lava flows. Most places, probably including the place where you live, have fossil bearing sedimentary rocks such as limestone, sandstone, and shale. These rock layers, however, may be covered by soil or plants, so you need a place where the rock layers are easy to reach.
Good places to find fossils are natural cliffs, or artificial cliffs in road cuts and quarries and at least shallow cliffs along a creek bank. The sediment rocks you are looking for are usually found in layers. Shale forms thin, flaky layers. Limestone is often white or grey and feels or marks like chalk. Sandstone feels more like sandpaper. Many cliffs have two or more of these types of layers.
Road cuts have been a great blessing for fossil hunters. Many newly cut roads have to be cut through many layers of sediment. Road cuts through mountains are excellent because of the rock layers they must go through to build the roads. Hunting fossils along public road cuts usually require no special permission. There is locations that do require special research permits such as national parks or public preserves. Hunting on private property requires permission of the landowner. Quarries are great fossil sites. These include gravel pits, stone quarries, mine tailings and shell pits. You need permission to hunt fossils in abandon quarries, and you need to be careful to avoid quicksand, rock falls, deepwater, and dangerous wildlife (such as alligators and poisonous snakes)
Tools needed for fossil hunting is containers such as bags, buckets, or boxes. You need notepads, camera, pen or pencil to record the location and time were you found the fossils. Other tools needed are shovels, hammers, brushes, shifters, etc.
There are techniques to fossil hunting such as using screening to find fossils. Wet screening techniques used in Florida can also be used for dry screening in drier environments such as in Arizona. When you find a fossil, there is ways to protect it using dampened pieces of paper (toilet paper or paleo-paper) with aluminum foil. Sometimes sheets of foil molded around the fossil makes a sufficient jacket without plaster. Other protection to fossils is using plaster-soaked burlap or other cloth.
The main thing needed is the fossil itself. The good news is that most fossils are easy to identify because things living today and fossils are both descendants of the same created kinds (Genesis 1:11-12,20-22, 24-25). Fossil clams look like living clams,fossil turtles like modern turtles. Every living thing that is alive today looks like its fossil counterpart. There is only a few major groups that are extinct today, such as dinosaurs that might be difficult to identify.
Fossils are categorized as petrification (which includes permineralization and replacement, preservation, carbonization, molds and casts, and trace fossils. Scientists such as paleontologists who study fossils know now that fossils need rapid burial which prevents decay and scavengers from destroying the bones.
Fossil hunting can be a fun time, looking for fossilized remains of creatures, such as dinosaurs. But it also is hard work that needs diligent dedication and patience in finding these fossils.