How much money do you want to spend?
Do you want a simple point and shoot?
Well it is a tough question.
From a high tech point of view you would have to move up to a super compact with the following :
-super fast zoom lens say from f2.8-f5.6 range for a clear sharp pic and other reasons
--a zoom of at least 10x to catch them in the field or indoors
-Camera must have a good flash to be able to hit more than 15 feet to light them up indoors and fill flash outdoors if needed
-Since they are moving you should have some sort of image stabilization to stop/ freeze the action
-Must have manual control settings such as colour saturation, contrast and sharpness, exposure..flash control etc
-should be able to have a really fast shuter speed range
-should have an ISO range at least 1600 for indoor shooting as the gymn light could be a combo of more than one type of lighting...otherwise the pic is going to be dark contrasted
-really important is it should have a burst mode and a continuous shooot feature to catch the action frame by frame
-most cameras now have a feature called skin tone detection as there will be some skin ( not mean to be lewd)
-It should have at least 6 megapixels
-it should have some sort of low light AF detector
You will be in a certain price range where all the competitors are equally priced and grouped with similiar features.
Brand names like Sony, Panasonic, Olympus, Fuji are good contenders..while Canon and Nikon have fallen behind with their money grabbing high prices and lack the zoom power with benefits that the competition has surpassed them.
AS an example I just gave away my Sony super subcompact Sony H-7 that could do all those things to my Father -in law and have bought the Sony DSLR alpha 200 with a collection of mid quality lens and a powerfull flash.
I did not mention a DSLR as your wife would not be able to handle one and learn quickly as most super compacts can be equal to the DSLRs with the kit lens options.
Check out a site called Steve's Digicams and check out the recommended grouping and you will see what I mean.
Go to a good camera store and get your hands on a charged loaded camera and test the results...as I visit and check out at least one super compact camera a week to keep abreast