@Bonehead is correct that galvanized mesh can be used where practical. It needs to be at least 18" high and 18" deep and should be bent so there is a significant lip on the gopher side of the mesh. I use the 1/4" mesh around my vegetable garden.
I've used traps with success. After trying a few types of trap I ended up using the Victor Black Box Gopher Trap but other people like other types of traps. Trapping is an art, you'll have to trap for a while before you get good at it. Luckily there is a lot of information on the 'net about how to trap gophers. With a trap my success rate is usually 1 out of 3, which means that if you have a lot of gophers you either need a lot of traps or be prepared to trap for a while to bring the population down. One advantage of traps is you know when you have killed the gopher.
I eventually settled on using poison as my main control method. That was a hard decision to make, since I don't like chemicals and I keep an organic, heirloom, non-gmo garden. But my success rate is much better and its the best way to keep the little beasties off my plants. Gopher Bait (poison) is applied down in the gopher tunnel and the gopher goes deep in the ground to his/her den to die, so there is little chance of a dog or cat or wild animal coming in contact with the bait.
Unfortunately you have to kill gophers to control them. The humane products that claim to make the gophers move away either don't work or only work for a short period of time. Even when they work they only move the gophers a short distance away, like into your neighbor's yard ("Thanks pal") and then they have babies who move back into your property.
However you handle them you are going to be faced with the issue of controlling the population in your general area. It's not enough to just get rid of the ones in your yard since there are gophers breeding like crazy just outside of your yard. Once you start to get your yard under control your little neighbors will move in. So you have to expand your efforts to include the gophers adjacent to your property. Think of it as creating a DMZ.
Good news: you probably have fewer gophers than you think. Gophers are solitary creatures, but they are very busy. A single gopher can have a 1/4 or 1/2 acre or larger territory but they can tear it up in no time. That usually means killing one gopher takes care of a large area or your property but you can't be sure until you get your control efforts underway.
You should also be aware that you have a mature tunnel system in your yard. All those nicely developed main tunnels and laterals are irresistible to a young gopher looking for a new home. So you are going to have to be extra vigilant for a few years until most of the tunnels collapse and its not as attractive.
Good luck and welcome to garden.org