Azamara is about the same as Oceania Cruises. These cruise lines are "niche" cruise lines that are between Premium and Luxury but are quite nice. However, while children are welcome they do not "cater" to children like the Premium and Mainstream cruise lines.
Off the top of my head I don't think they have a children's program on board. I may be wrong about this and someone can feel free to correct me.
I've sailed both lines, but it's been a long time since I cruised Oceania, and I definitely prefer Azamara. The food was far superior and the service was much better.
The prices will be more than most Mainstream (Carnival, NCL, Royal Caribbean) and Premium (Celebrity, HAL, Princess) cruise lines but usually less than a Luxury line. (Regent Seven Seas, Seabourn, Silverseas)