best advice you would give to a new (used) concours owner

[Two Skies]

Well, besides investing in overflow tubes or some other measure to deal with the dreaded hydrolock (harbinger of death for Connies), I'd suggest looking seriously at tipover bars.

I put off getting them for my bike, and thanks to a bad day, and in an unrelated incident, a drunk person knocking my bike over and falling on top of it, as well as some other minor incidents I have some significant damage to the plastics and pegs.  Not to mention needing to repair the sidebag brackets as well.  Oh, and of course the mirrors.  If you want your bike to stay pretty and avoid the potential for some somewhat costly repairs if she ever goes over, invest in tipover bars.  These won't be 100% foolproof, but will give you peace of mind r.e. minor incidents.

http://www.murphskits.com/catalog/product_info.php?products_id=127

[roadkoan]

+1 for the tip-over bars.
Cheaper than new plastics, and I think they look cool too. (just IMO)

And most importantly, buy good tires!
I'm not just talking about ones with tread either. Unlike men, not all rubber is created equal, and when I went from the Bridgestones that were on her when I bought her to the Avons I ride now, it was like night and day. Maybe they wear out sooner being softer, but when the chips are down there is no better insurance than good traction! 

Oh, and practice, practice, practice. I get cone patterns here Go cut some tennis balls in half to mark the course and don't forget to have some fun!

[Boomer]

FONDLE! Then Fondle some more.
Ride, then Fondle.
Fondle in the privacy of your own garage.
Fondle, fondle, fondle! 

My personal standout fondles were;
Fondled Jerrys bike in a motel car park in Klamath Falls. Gene was a fellow fondler and much beer was drunk.
Pic of Jerry and me is here http://gtr1000.com/graphics/boomerpics/Boomer_Fixin_Jerrys_ride.jpg
Fondled Hans-Oves bike in Denmark @ first GCE Rally.
Fondled Walters bike in Altenmarkt, Austria after he binned it in 2003.