All quotes, etc., in this essay were transcribed from my rewatching of the DVDs.
"The fun is just beginning." --Rupert Giles, "The Harvest."
Giles doesn't make his first appearance in "Welcome to the Hellmouth" until about ten minutes into the episode; he's the last of the Core Four to be introduced, and the only (major) character introduced later is Angel. It's fitting, really, because season one of Buffy the Vampire Slayer is very much about high school, and adults, even Buffy's Watcher, have to take a position on the sidelines.
Giles is, of course, professionally on the sidelines as Buffy's Watcher, as well as being outside the usual sphere of Buffy's daily life as a high school student. He takes a less central role in the first season than he will later (the second season, for example, gave him a tragic romance and supplied some of his back-story), and while there are episodes that shine a spotlight on Willow ("I Robot... You Jane") and Xander ("The Pack"), there are no Giles-centric episodes in this season.
Giles may start out on the sidelines in the first season, but over the course of the twelve episodes, he's pulled closer to the center of things, professionally as well as personally.
"It's a welcome change to have someone else explain these things." --Rupert Giles, "The Puppet Show"
Giles is, of course, in Sunnydale as Buffy's Watcher, and his enthusiasm for his job during the first season is in sharp contrast to Buffy's frustration and attempts to reject her destiny. From his first scene, as he happily brings forth the huge "VAMPYR" book for Buffy, it's obvious that he's excited to be in Sunnydale, to be Watcher to the Slayer, to be able to witness the odd mystical events surrounding a Hellmouth. (He even bought the Time-Life books, after all.)
We learn from Giles that the Watcher's role is to train the Slayer, to prepare her for the evils that she must face, and indeed, we do see him training Buffy several times this season. His other main role is, apparently, research, and the majority of Giles' scenes in the first season do take place in the library. Giles is used by the writers to deliver large blocks of exposition and to refresh viewers' memories about ongoing plot threads (in "Teacher's Pet," for example, he rather unnecessarily describes the Master as "our local vampire king"), to the point that when, in "The Puppet Show," he gives the line quoted above in response to Sid's explanation about the demon, the viewer may very well agree: it is a relief to let someone else be Exposition Guy for an episode.
At the first of the season, it seems that Buffy's Watcher is best qualified to... watch. When Buffy sees Willow with the vampire at the Bronze in "Welcome to the Hellmouth," Giles immediately asks her what they should do, and whether or not she needs him to come along with her--obviously at least half-hoping she'll say no. He may have the training and the knowledge, but he doesn't seem at all confident at the thought of putting that into practice. By the next episode, however, we see that he's willing to jump in where needed; he takes part in the final battle at the Bronze without hesitating.
From that point on, we don't see that same uncertainty in him; he may not always be the most effective man in a physical fight, but he shows no hesitation in doing whatever is necessary to help Buffy. Because Buffy is the Slayer, of course, Giles' help is often less physical and more oriented toward research and magic, but it's important just the same--for example, while Giles claims in "The Witch" that the spell he performs to return Amy and her mother to their own bodies is his first casting (though after viewing later seasons, the audience may doubt his truthfulness), the magic he works is instrumental to the plot in both "The Witch" and "I Robot... You Jane." He may be a Watcher, but he's obviously redefined his role to include far more than just training and preparation.
And, in "Prophecy Girl," he redefines it yet again when he's prepared to face the Master in Buffy's place. He's not training his Slayer, he's not preparing her for battle, he's not providing support; he's intending to take her place in the hope that it will save her life--and knowing that it's very unlikely that he'll survive. This is no longer an intriguing intellectual problem; this is the life of a girl he's come to care about. Enthusiasm has been replaced by determination, his faith in his reference books has become defiance of prophecy--and Giles, who's meant to be watching from the sidelines, plans to sacrifice himself in Buffy's place. It's a far cry from his hesitation in "Welcome to the Hellmouth," that's for certain.
"I'd say he should get a girlfriend if he wasn't so old." --Buffy Summers, "The Witch"
Of course, one of the reasons for Giles' willingness to sacrifice himself for Buffy is that he's gotten to know her over the past few months; his unconventional Slayer has made him interact with her, and her friends, on a personal rather than a strictly professional level. In "Welcome to the Hellmouth," nothing seems less likely; his interactions are with Buffy, and when he goes to the Bronze to look for her, he admits that his idea of a good evening is being at home with a cup of Bovril and a book. It's not surprising that a teen hangout isn't Giles' favorite place to be, of course, but his solitary choice of activity just emphasizes that he's not yet ready to interact with the world he's found himself in.
Ready or not, Buffy gives him very little choice. When he discovers that Willow and Xander know that Buffy is the Slayer, he gives in quickly to the inevitable and puts them right to work as his research assistants. He's expected to cope with normal teenage behavior despite--as he says--not liking teenagers very much. While that might be true in general, (certainly, his "he's becoming a sixteen-year-old boy" speech to Buffy in "The Pack" seems to support it), he does seem to care about these three teenagers, most particularly Buffy.
Even as early as "The Witch," it's obvious that he's begun to care about her; his treatment of her while she's ill from the "bloodstone vengeance spell" makes that clear. By "Nightmares," of course, he's become very genuinely attached to her; his speech at Buffy's grave toward the end of that episode is laden with strong, if quiet, emotion. "Prophecy Girl" I've already mentioned, of course; a Watcher's traditional duty doesn't extend that far, not from any canonical indications, and so there can be no other explanation than that Giles has become emotionally involved--he can't stay on the sidelines any longer; he has to act to save someone he cares about.
While Giles' relationship with the other two is, of course, neither canonically as close nor the focus of much attention, we do see him with them throughout the season, and not only while fighting monsters, and their relationship is comfortable enough by "The Puppet Show" that the three younger Scoobies feel able to tease him about being put in charge of the talent show--and, in Xander's case, to give him tips on how to deal with Cordelia.
It's not just with Buffy, Willow, and Xander that Giles has had to become more involved as the season progresses. He has to learn to regard Angel as an ally, despite his past as Angelus. That's made far easier by Angel's offer of the Pergamum Codex, of course, but he does seem to have genuine sympathy for Angel's situation with respect to Buffy.
And, of course, no discussion of Giles' personal relationships in season one would be complete without mentioning Jenny Calendar. We know as early as "Teacher's Pet" that no matter what Buffy might think, Giles isn't too old to notice an attractive woman. His relationship with Jenny gets off to a very prickly start, though; the two of them argue throughout most of Jenny's scenes in this season. He does, however, gradually come to appreciate that she might have something to offer the group, and even trusts her with Buffy's secret. The romance of season two has not yet begun, but the foundation has been laid; he's made the decision to trust her, to involve another person in what's going on. It's a far cry from the Giles of "Welcome to the Hellmouth," who'd have preferred that Buffy kept her secret from everyone.
The final scene of "Prophecy Girl" illustrates best, I think, how far Giles has come from those days; when everyone--Jenny included--decides to go to the Bronze to celebrate Buffy's victory over the Master, Giles doesn't protest, going along with them and only insisting that he isn't going to dance.
Willingly or not, he's become part of the group. He's still separate from Buffy, Willow, and Xander, as he has to be, given his role as the token adult--but he's not standing on the balcony at the Bronze, watching the rest of the world and wishing he were home with a good book, either.