People have been fishing about as long as they have been hunting. People began to fish before the first people left Africa, about 60,000 years ago. Because people in the Stone Age were fishing for food, not for sport, they did not feel bad about using easy ways to catch as many fish as possible as quickly as they could. Mostly they used nets.
You could make nets by spinning grasses, or spinning wool, or flax. Often men did the spinning to make nets for fishing or hunting. Then you knotted the strings together in complicated patterns like macrame to make the nets, and you attached stones or bits of clay or lead to the bottom of the net to weight it down (so it wouldn't float), and you attached bits of wood to the top (so it would float).
A group of people would wade out into the ocean someplace where it was pretty shallow, and when they were out as far as the fish they would spread out the net, and walk forward, and then the men on the end would walk towards each other and close up the net, and they would all carry it back to the shore.
If the water was deeper, you could also fish from small wooden boats, by dragging the net behind you for a while and then pulling it up into the boat. This is what Peter was doing when he first met Jesus, and that is what the people are doing in the Roman mosaic at the top of this page.
By the time of Jesus, though, people had fished so much around the Mediterranean that there was beginning to be a shortage of fish. To make up for this, people were already farming a lot of fish in artificial fish-ponds.
Learn by doing: Go fishing! (but be sure to get permits)
Starting Fishing, by Fiona Patchett and others (1999). For kids.
Kids Book of Fishing/Book and Tackle, by Michael Rosen (1991). Includes fishing line, hook, and so on, in case you want to try it out for yourself.
Cod: A Biography of the Fish That Changed the World, by Mark Kurlansky (1998). For adults, mainly for entertainment, but it gives a good idea of what fishing was like in ancient times.
Fish and Fishing in Ancient Egypt, by Douglas Brewer and Renee Friedman (1990). By specialists. Includes a history of fishing, and also helps archaeologists identify fish bones.
Piscinae: Artificial Fishponds in Roman Italy, by James Higginbotham (1997). By a specialist, for specialists.