So if you have to switch, how do you make it work?
If the new rink is going to be your new home rink, work yourself into the culture there, but not too fast. First step is:
Hang out for a week--drop in and find out which sessions are crowded, which tend to be "high" and which have more beginners. Watch the classes to see what coaches interact a lot with the kids, and if there are any that make a point of coming off the ice between classes (if they can) to talk to parents. Are there classes where all the kids are smiling (or classes where they're not)? When is the ice show or exhibition, and what are the rules for participating (some rinks put up barriers to this, like you have to be in the program for the prior two sessions to get a solo, which is insane-- what about the Freestyle 8 kid who just moved to town? Sorry, no, you're not one of us.)
Second step is a negative one:
You don't know anything about this rink. Don't come in talking about how your other rink did xyz differently; that statement will be interpreted as "you guys don't do this right." Absorb the culture, don't roll over it. I would frame all my initial interactions as a search for information: "we're new here, how does this rink do [whatever]?" NOT "I don't get how you do this here, at our old rink, it works like this; blah blah blah." (That is what people will hear. We're basically a selfish species, we don't care how it works at your old rink, we're never going there, especially if everyone is like you.) Don't bash the old rink or the old coach or any skaters you knew there. Figure skating is a very small world. Forget 6 degrees of separation. In the skating world, it's more like 2. I can get to pretty much any skater you can name through just another two people . Further, this is how skaters talk to each other when they first meet. They'll determine where you're from, then start with, oh do you know Coach A? Well, no, but I know her student Y, etc.
Snark at the rink will come back and bite you on your frozen arse.
Third step is get involved
Sign your skater up for a class. Sign up for two! Take a class yourself. If it's ice show time, volunteer. Sign your skater up for the ice show. If you've missed the deadline, talk to the skating director and see if there's a way for your skater to be involved (last year two of our best skaters had to compete the week before the show and couldn't be in it, so we found them a couple of old costumes and sent them out as tot wranglers). Introduce yourself to the moms and dads in the stands (remember, no old rink gossip or "we do things better!") Bring a cake for the next birthday or holiday, set it out in the lobby for all the kids. If there's no holiday, just make one up.
There are new families at our rink who you'd swear have been there since the kids were tiny, because they've absorbed the culture and made themselves a part of it. And there are kids who grew up there that still sit on the sidelines, looking forlorn and lonely, because their moms won't talk to anyone, won't sign them up for classes, and yank them away the second they're done.
Guess which kids will keep skating?