This is definitely a personal style kind of choice. Here's how I do it. I am not a very touchy-feely SM, but I do like making friends on shows and staying in touch with them. The 700 people in my computer address book are evidence of that, lol! The computer really is what makes staying in touch possible. Certainly, on any given show, only a handful of people are added to my close group of friends who I might idly email once a month, or get a drink with if they live near me. Beyond that, as I add people to my address book, I put them in my "Xmas Email" category, and that way I'm sure to add them to my list when I send out my yearly update email of what I've been up to. I think we all know that it's impossible to keep in close touch wioth everyone in this business cause we meet way too many people. But we also know we'll likely get a chance to see each other again. I just try to pay attention to who's where at any given time, and if I ever pass thru a city where I know some old theatre friends are at, I'll give them a call.
On the tour itself, I think everyone finds their little groups to hang out with, and if the tour's long enough, those groups will likely change over time. I'm a bit more laid back about the excitement of seeing people at sound check every week. I mean, the cast hasn't seen eachother for almost two days, but I've been their with the crew since earlier in the day, so it's not as big a deal to me. Nevertheless, there's always a couple people who I'm genuinely happy to see and whom I might make a point to go say hi. But that all develops as the friendships do. I certainly wouldn't buy people gifts at every venue, just like I don't buy my roommates back home a gift every week. I just treat friends on tour like I would real-world friends. And for me, it is important to be friendly and relax with my friends on the tour. I need a life and that's what keeps me sane, by having good friends. Mind you, I tend to avoid the big company-wide, group-hug, kumbaya parties, cause I'm NOT close friends with all of them. But I will generally at least make an appearrance at these events.
And yeah, it is hard to leave the tour after it's over. After my last 5-month tour ended, for the next couple weeks back in NYC, several of us stayed in touch almost daily. We just couldn't bear to be apart. But eventually you ease back into your normal routine, made all the richer for the addition of some new friends!