They are usually pretty timid and cryptic at introduction, and they slowly become bolder with time. Personality wise, they are pretty similar to a Harlequin Tusk, maybe not quite as active of swimmer, but a larger fish overall. As with most large wrasses, if the tank is too small, they'll hide most of the time. Keep in mind the Cheilinus genus of wrasses, also includes the Humphead/Napoleon wrasse, one of the largest reef fish on the planet. Red Breasts stay smaller then Broomtails or TripleTails (the other two common Cheilinus besides the Snooty wrasse), but still get 15", and unlike a Lunare or Sunset, they are much more heavy bodied, no so torpedo, more like a Snapper.
I have read a few large Cheilinus's that have been terrors, so try to look for a younger fish or a smaller adult. Tankmates are kind of tough, at intro, they can easily be bullied, but anything too small might be a meal. Snappers, Puffers, Hinds, etc, make good tankmates, generally fish that keep to themselves. They usually do fine with other species of wrasses, but care ought to be given if attempting.
I believe 95harley had a Broomtail wrasse. If he doesn't response PM him here, or on the another fish board.