Enterprise was the series that explored the Vulcans more than any of the other series in the Star Trek Universe. The character of T'Pol was central to many of these stories. Another character that made a frequent appearance was Soval, the Vulcan ambassador to Earth.
The Vulcans have a relationship with Earth that is a mix of co-operation and adversarial emotions. All through the series, we see that Vulcan society is going through a revolution, with things coming to a head in the final season of the series.
The Vulcans are in a long term conflict with the Andorians and several episodes feature that conflict and how Enterprise was involved in them.
1x07 - The Andorian Incident
Enterprise visits the Vulcan monastery P'Jem, only to find that it has been taken over by the Andorians.
1x08 - Breaking the Ice
With a Vulcan ship on their tail, the Enterprise decides to investigate a comet.
1x17 - Fusion
The crew of Enterprise meets a group of outcast Vulcans, who embrace their own emotions.
1x23 - Fallen Hero
Archer is ordered to transport Vulcan Ambassador V'Lar from the planet Mazar, where she has been accused of criminal misconduct. Following an attack on the Enterprise, T'Pol defends the Ambassador and asks Archer for help.
2x02 - Carbon Creek
T'Pol tells Archer and Tucker a story about her great-grandmother and two other Vulcans, who crash landed in a small Pennsylvania town in the year 1957.
2x14 - Stigma
T'Pol's position on Enterprise is jeopardized when the Vulcans discover that she has contracted a mind-affecting disease.
2x15 - Cease Fire
Archer is called in to mediate a territorial dispute between the Vulcans and the Andorians.
3x05 - Impulse
After finding the drifting hulk of a Vulcan cruiser inside an asteroid field, Captain Archer and his crew are surprised to find zombie-like Vulcans on it. When they are stranded there, they must find out what happened to the crew before T'Pol succumbs to the same fate.
4x03 - Home
Tucker visits the home of T'Pol on Vulcan and meets with her mother. T'Pol makes a decision regarding her engagement. These four episodes form a storyline that follows a very critical part of Vulcan history and Captain Archer's part in it.
4x07 - The Forge
When Earth's embassy on Vulcan is bombed, the investigation puts Archer and T'Pol on the trail of a Vulcan religious faction hiding in a treacherous desert.
4x08 - The Awakening
Archer and T'Pol encounter the Syrrannites, the radical group supposedly responsible for a terrorist bombing on Vulcan.
4x09 - Kir'shara
Archer and T'Pol try to recover a Vulcan artifact critical to Vulcan history and society. Enterprise gets involved with a standoff between the Vulcan and Andorian fleets.
The character of Spock is an iconic one in pop culture. While he was ever present in the original series, there weren't too many episodes that were dedicated to him personally.
2x05 - Amok Time
Suffering through his first infliction of pon farr, the Vulcan biological mating urge, Spock must return to Vulcan to marry his betrothed or he will die. However, when the Enterprise arrives at Vulcan, complications at the ceremony may endanger Captain Kirk as well.
2x15 - Journey to Babel
As the Enterprise comes under attack on the way to a diplomatic conference on Babel, one of the alien dignitaries is murdered, and Spock's estranged father Sarek is the prime suspect--but he is also deathly ill, and only Spock can save him.
3x04 - The Enterprise Incident
Kirk inexplicably orders the Enterprise across the Romulan Neutral Zone, where it is quickly surrounded by the Romulans. Kirk and Spock are brought aboard the Romulan vessel where the female Romulan commander attempts to persuade Spock to side with his Romulan "cousins," take command of the Enterprise and accompany the commander back to Romulus.
3x06 - Spock's Brain
The Enterprise is raided by an alien force that steals Spock's brain, leading Kirk and McCoy in a desperate race to retrieve it.
The Next Generation was surprisingly low on Vulcan characters and story lines. There was no major Vulcan character on the crew of the Enterprise. The token non-human was a Klingon.
3x23 - Sarek
Ambassador Sarek of Vulcan is called aboard the USS Enterprise-D to fulfill his latest diplomatic duty, a treaty with a mysterious race known as the Legarans.
5x07, 5x08 - Unification
To the Federation's surprise, Ambassador Spock has traveled to Romulus. Fearing he has defected, they send Captain Picard and some of his officers on a covert mission to determine why.