Just fyi, the delay is the camera focusing. You cannot get rid of this unless you get a professional camera that you manually focus. There are some cameras that do it faster and some that do it slower, but any SLR camera is going to do it.
Another thing that is good to look for is a high optical zoom. The higher this is the better you can zoom in and get clear pictures. Digital zoom makes everything pixel-y and is useless in my opinion. Megapixels are really just a buzz word. Unless you are going to be blowing up your pictures to gigantic poster sizes, and modern digital camera really has plenty of megapixels.
I really like my camera...it is a Canon but it is an older model that is discontinued. I have tried Sonys as well but the flash always seemed really harsh to me...kids looked like they were in shock in all the pictures.
I would look up digital cameras online and then write down model numbers that look interesting to you and try to find sites that have reviews of them. Newegg.com is a good place to look, always has good prices and lots of reviews.