There are many techniques to catch wild fish. Hopefully this page serves as a repository for cheap techniques.
Limpets can be gathered freely in most rocky shores in saltwater. Most people use a knife to remove them but a thin rock will do. Just hit them close and fast and they will just dislodge from the rock.
Catching fish in the wild with minimal equipment: http://www.instructables.com/id/Building-A-Cheap-Fish-Trap-DIY-For-catching-fish/
Angling is perhaps the most easily-recognized form of fishing. It involves using reels, rods, and bait or lures. Anglers usually fish from boats, but can fish from the shore as well.
Ice fishing is a wintertime variant of angling involving fishing through a hole cut in ice on the surface of a body of water.
Fly fishing involves casting elaborate lures on lightweight line, usually in shallow streams or rivers.
Bowfishing involves catching larger fish using archery equipment.
Clam digging is digging clams out of the muddy intertidal zone that forms their habitat. Clams can be located by bubbles formed by their respiration.
Bait and Lures
Fishermen use bait and/or lures to entice fish to strike.
Lures are manufactured objects intended to resemble swimming prey fish. They are designed to spin or wiggle when drawn through water, so as to resemble the motion of small fish. They have the benefits of being reusable almost indefinitely and never spoiling, but the drawback that they must usually be purchased.
Live worms, insects, minnows (small fish), frogs, and other small aquatic creatures can be used as bait. They may be collected by hand or purchased.
Minnows can be harvested from streams using a minnow trap.
Earthworms can be cultivated. A home vermicompost operation can produce surplus worms that may be used as fishing bait.
Scrap meat can also be used as bait: leftover lunch meat, trimmings, or offal from previously-caught fish.
The process of butchering edible fish is called cleaning.
Cleaning requires a knife. Just about any sharp knife can work, although there are fillet knives and descaling knives designed for those specific purposes.
See: Cleaning Your Fish
Fishing is regulated in many areas. Many US states require a fishing license, which typically involves a modest fee and no other requirements.
Many bodies of water have restrictions on the number of fish that may be caught, sometimes on the basis of specific species. These rules are usually either intended to preserve a threatened species, or in the case of artificially stocked bodies of water, distribute the stock fairly.
Some techniques may be banned or limited; for example nets and bows are often regulated or banned.