Thereis a significant delay between pushing the button and the pi

This is a common situation with almost all point-and-shoot cameras. The delay is because the camera has to do so much when you push the button. Bulkier and more expensive SLRs eliminate the delay by having more hardware to handle the various tasks.

You can reduce the delay by anticipating the action. If you know where the action is going to happen (a child blowing out the candles on a cake, or right in front of a soccer goal, for example) aim the camera there and press the shutter button halfway and hold it there. This meters the exposure and focuses the lens. Then when the action finally happens, press the shutter button the rest of the way. With the camera having done most of the work when you pressed the button halfway, there will be much less delay.

Again, the delay is a basic "feature" of the camera design. It can't be completely eliminated, but by pressing the shutter release halfway it can be reduced.

Are your pictures properly exposed and are they sharp or are your pictures degraded? Is that 1/3-second you describe a delay between the time you push the shutter release and the time the picture is taken or somthing else? Check to make sure you're in the proper shutter release mode (slider switch to right of the mode dial). Also, your self-timer may be turned on. Your camera allows you to set a 2 or 10 second delay from the menus. If it is set to 2 seconds and turned on, it might seem like a 1/3-second delay. If that doesn't help, you camera is covered by Nikon's warranty. Contact Nikon Service at 1-800-NIKON-US (1-800-645-6687) 9AM-8PM EST, Monday to Friday.

"R(x)" or "R(xx) is the estimated numbeer of shots left on your storage card. For the delay: with camera in the "ON" position, remove battery(ies), and while waiting approximately 30 seconds, continue holding the power button. REplace batteries and see if that helps. Also, if your model has a hardware reset button, give that a go.

Shutter lag is a universal problem with digital cameras...more so with the point-and-shoots than the slr types. To avoid the problem, press the shutter button half-way to lock in the focus then press the rest of the way when the moment is right and you will get no shutter lag.

are you using mirror lock-up or live-view feature?

It is probably delaying while it autofocuses.  If it is like most Canon cameras, depressing the shutter half-way should allow you to focus without taking a picture, then when the moment is right, depress the shutter button fully to take the shot.  You will want to practice this a bit to get used to it.  Shooting in low-light it can take longer to find the focus, so it becomes more necessary to use this technique.
Hopefully that helps, but I am not a Nikon user.

I'm guessing you have a delay set (people will use this on a tripod). The small button on the top left (next to the flash control) will solve your problem I think.

Are you sure you aren't in the delay setting in drive mode? This camera has 2 delay settings, one for 2 seconds between when you press the shutter and when it takes the photo, and one with 10 seconds delay - often used when you want to be in the photo (e.g. self-portrait or group photo).

If you aren't in the delay mode, then I need to know more about your settings. What shooting mode are you using? What type of photo are you trying to take (portrait, landscape, sports)? Are you shooting indoors, outdoors, bright sunlight, overcast, etc.?

I also have this problem on a Canon A85. For some reason setting the camera to landscape allows the shutter and flash to work together properly.

Melzim, another user, is exactly right. The lower-end digital cameras experience a significant shutter delay when using the camera; it's an unfortunate fact. 
What you can do, outside of investing in a newer model (which do have improvements over the earlier models) is simply try to anticipate your subject's movement - or concentrate on portraiture, etc. 
There is a huge value, over time, in having pictures of smiling families, standing still, believe it or not. 
The A85 is simply not the camera to use to freeze action.
-lizwb