Note: When Civil War historians talk about the "Western Theater" of the American Civil War, they're usually referring to the campaigning that happens east of the Mississippi River and west of the Appalachian Mountains.
What you are asking about is the Trans-Mississippi Theater. This theater of the war encompassed all the fighting west of the Mississippi River.
Before I go any further, I'd like to recommend this booklet by the U.S. Army's ever-helpful Center of Military history. It has the advantages of being free, brief, and digestible in one sitting. It also has quite a nice selection of maps, and I'm a sucker for a good-map.
The conflict in the American West was quite active. It was extremely interesting, with a succession of stagecoach robberies, bungled attempts at piracy, sharp skirmishing, and hot pursuits. It was also pretty inconsequential in the grand scheme of things. Of course, it was very consequential to the people who lost lives, limbs, and livelihoods. At the end of the day though, the war wasn't won or lost in California or New Mexico.
Let's use Gordon Chappell's summary of the theater as a starting point:
[The] New Mexico Territory was an active theater of the war with two major battles and a number of lesser engagements, the Gulf Coast of Texas was under Union assault periodically, Confederate guerrillas operated in Colorado Territory, and Confederate soldiers occupied the cities of Albuquerque and Santa Fe in New Mexico and Tucson in Arizona. The westernmost combat in the war occurred south of Phoenix in what now is Arizona at Picacho Passwhen a detachment of Confederate Texas Mounted Riflemen ambushed a patrol of the Union 1st California Volunteer Cavalry, and an even smaller skirmish between Union and Confederate soldiers occurred still farther west, at Stanwix Station, only 80 miles or so from the border of California.
In 1865, the Confederate States Navy commerce raider C.S.S. Shenandoah destroyed part of the civilian Union whaling fleet in north Pacific waters, scaring the living daylights out of the populace of San Francisco, which feared they and their banks full of gold from the Mother Lode mines would be raided next.
So there you have several dimensions of the war in the American West. There's a handful of open battles. They're small by comparison to the massive multi-day slogging matches of Gettysburg or Shiloh, but they were significant given the context of the fighting. For a small army, even a small battle can be a very high-stake affair. There's the guerrilla warfare, raiding, and out-and-out highway robbery. And there's the oft-forgotten naval campaigns of would-be raiders and privateers.
Since Chappell mentioned New Mexico first, I figure that's a good place to start. The New Mexico Campaign of 1862 was a daring effort by Confederate General Henry H. Sibley to take the war west.
Sibley and his mustache had an ambitious plan. With a few thousand men, Sibley would take Colorado Territory. This would give the Confederates control of Colorado's mineral wealth and Fort Laramie. Oregon Trail players will recognise this as a key stopover on the trail. From there, Sibley planned to march westward into Nevada, California. This would better link the newly-seceded Terrority of Arizona with the rest of the Confederacy. He even had designs on Mexico. If the Mexicans didn't let Sibley buy territory with dollars, he'd take it with lead.
To top it all off, Sibley thought he could accomplish all this while packing light. Captured Union supplies would provisions his soon-to-be victorious troops, Sibley assumed. In fairness, Sibley had been on a winning streak up to that point. He'd captured Santa Fe and Albuquerque and won a victory at Valverde earlier in 1862.
Sibley's campaign was rather unceremoniously ended on March 26-28, 1862 at Glorieta Pass. In the largest battle of the Trans-Mississippi Theater, Union troops forced Sibley's army to retreat. Unsurprisingly short on supplies, the Confederates had to turn back and call off their campaign.
Later that year, Confederates enjoyed better luck. At the April 15, 1862 Battle of Picacho Peak, Confederates turned back a column of pro-Union troops from California trying to retake Tuscon. With just over 20 men involved, it was a small battle. Still, still it was the westernmost battle of the Civil War.
Although it didn't see any major fighting, California was actively-involved in the war. Pro-successionist Californians robbed stagecoaches to raise funds for their cause. There was one abortive attempt to send out a privateer and another failed attempt to hijack a ship for similar purposes.
Many Californians also headed east to fight, as Chappell notes:
...an unknown number of individual western civilians traveled back east to fight in either the Union or Confederate armies. The one well-known example of a unified effort in that direction was the group of 504 Californians who did not want to spend the war fighting Indians out West, so traveled to Massachusetts at their own cost and persuaded the State of Massachusetts to enlist them under its quota for volunteers in five companies of the 2nd Massachusetts Volunteer Cavalry. One group of them came to be known as the “California Hundred” and wore atop their forage caps in addition to the brass crossed saber cavalry insignia, the brass letters and numbers “C A L 1 0 0”.
Confederate sympathizers had to be chased down and captured before they could join the Rebel army. This lead to some dramatic chases across Southern California, as pro-Union volunteers pursued would-be Rebels.
Of course, there were also long-running skirmishes and clashes with Native Americans. Some, like Cherokee Stand Waitie fought for the Confederacy. Others fought for themselves and their tribe.
Some of the American troops sent to defend and expand the frontier were rather unlikely soldiers. They were the so-called "Galvanized Yankees," former Rebels who had traded in their grays and butternut for blues.