You've received many excellent recommendations. Add to them Tony Parkes' dance collection books --- Shadrack's Delight and (I think) Son of Shadrack. They include squares that are highly contra dancer accessible. And since Tony can't toot his own horn, do get his book on square calling and do attend any dances he calls if possible. He danced on Long Island to  Ed Durlacher's calls --- author of Honor Your Partner and one of the leading lights of the huge post WWII square dance explosion. This was a pre-television era when square dances were staged in Central Park and thousands of people participated. He's your finest living link in this great tradition. (Pete met Toshi Seeger at the square dance in NYC).

Ted Sannella, in addition to starting the Ralph Page Legacy weekend, included the idea of publishing a syllabus with all the dances included. These syllabi, maintained over the years by David Smukler, are an extraordinary resource and include lots and lots of great square dance material/info. All are available online via the University of New Hampshire Library. (google search will find them)

Organize Dare to be Square: NYC and build your own caller education track into it.

 

"Laugh when you make a mistake," John Krumm circa 1995. Have fun and share your joy with your dancers. If anything goes wrong, roll it in a tight little ball and tuck it away in a pocket somewhere to think about later. Emotions are contagious. You want to spread happiness.

When you call squares, you're part of the band. This is super true with singers, but also when calling any square. You're the lead instrument. Your sound people should know to dial back the band and keep you in front in the mix. 

No cards, phones, tablets, or other distractions allowed while you're calling a square. Know your material cold. Keep your attention reserved for your dancers and your musicians.

When calling for contra dancers, call to the music--not the dancers. If a square falls apart, don't watch it -- you can't fix it on the fly. If things go crazy all over the floor, laugh, square 'em up, and start up again.  

On the dance history project site, do watch Tony calling the Merry Go Round.  

Many of Ted's dances can be tricky for contra dancers and tricky to teach. But Joyeux Quadrille is not.  You can dance it to the name tune per Ted's suggestion (a jig, btw), but done to southern reels with energy, it can bring down the house. Tony does a neat change-up that's easy to accomplish with the long lines to corner swing transition, alternating between ladies and gents with the allemandes (and you can throw other stuff in there--an opportunity for fun).

Tony's dances -- for starters, Star Breakdown, Left-Hand Star Breakdown, and Duck Through and Swing

Contra dancers like keepers -- where you end each time through the figure with your original partner. 

Tom Hinds: Shooting Stars

Gene Hubert: Kimmswick Express (32-bar version) and Early Autumn Express (32-bar version)

Bob Isaacs is making up come contra dancer accessible squares these days. And he has some grid squares that are quite manageable when you have dancers who can handle all the basics. (And a shout-out to Kathy Anderson who was calling grid square at the Ralph Page Legacy Weekend back in the late '90s.)

For graduate studies, explore The Teakettle, created by Ron Buchanan.

Get comfortable with doing Appalachian-style squares where you do a visiting couples figure like Rights Hands Across or Duck for the Oyster. Do the figure, then it's "circle up four, find another couple and circle some more." These are dances you can call for dance parties, contra dancers, and anywhere in between. You can start one up and people can join at any time. Visiting couples figures can be boring in a four-couple square and a lot more fun in a big circle. 

And laugh when you make a mistake. Have fun.

R