There is nothing wrong with using P mode from time to time.... Sometimes it's the best mode to use, in fact.
What metering mode did you have set (Evaluative, Center Weighted, Partial, Spot), what ISO and what were you pointing the camera toward? Any filters on the lens?
Assuming a normal, sunny scene that averages out to around 18% gray, it does sound as if the camera is reading low... at least four stops off. Even ISO 100 in full sun should end up at 1/400 or so in full sun at f5.6.
However, the readout often varies wildly in any of the AE modes, isn't what's actually used to take the shot. Were images poorly exposed (way, way overexposed), or are these just the readings you were seeing displayed (either in the viewfinder or on the LCD)? Are these the settings recorded in the image(s) EXIF metadata?
Try Tv, Av and M, too. If they work differently, there may be a problem with the camera. However, I'd expect the problem to carry over to different lenses and you say another lens you tried seemed to work okay.
Another thing you might try is setting the camera to the "Green Box" mode.... That's sort of like Program on steroids, won't allow you to shoot RAW and will probably limit metering modes and AF modes, but might help identify a problem.
Inspect the mirror and inside the mirror box carefully, behind the mirror by lifting it gently with a toothpick or similar. Any obstructions? The most likely thing is a piece of light seal foam breaking off and partially obscuring the metering sensors.
I think it most likely that there is a metering system and/or AE system problem in the camera, however the fact that one lens works okay while another gives problems doesn't support this.
Usually it wouldn't be an issue with the lens causing these types of problems.... More commonly a lens will have a stuck aperture, but that wouldn't effect what the camera is trying to use and displaying, the lens simply wouldn't stop down properly. AF motor failure is another most likely thing to happen with a lens, but would be unrelated to exposure. Wear and tear of the mechanisms are another typical problem with lenses, but also usually wouldn't effect exposure.
However, you might try cleaning the contacts between camera and lens, to be sure they are communicating properly. Just use some isopropyl (rubbing) alcohol, a couple drops to dampen a clean cloth, and wipe with that. The most common problem effecting contacts are oils, either from fingers or lubrication on the camera's mechanisms or just from the plastics used in the camera. For that reason I don't recommend using a pencil eraser on the contacts... those are made from vegetable oils.
Also "reboot" the camera. It's possible some sort of garbled instructions are "stuck" in the circuitry. It's a lot like rebooting a computer to clear RAM. To reboot the camera first turn it off, then remove all the batteries - both the main rechargeable and the little silver memory battery - then turn the camera back on and press the shutter release button once. It won't trip the shutter, but that should drain all remaining power from the camera, forcing a "reboot" of sorts. Turn the camera off, reinstall the batteries and turn it back on... then check the date/time... they should need to be reset if you got a proper reboot. If not, you might need to use the longer reboot process... same as above except don't bother pressing the shutter button... instead leave all the batteries out for at least an hour, perhaps even overnight... then reinstall and check the date/time. You might need to reset some menu selections or custom functions you had set.
You also might try flashing the Firmware in the camera, even if you are up to date with the latest. Just download the latest version from the Canon website and carefully follow the instructions you'll find there. It's not difficult, but you want to be sure to do a sum check (or is it a check sum?) of the FIR file to be sure you got a complete download, and you have to be careful to not turn the camera off or bump any buttons while the firmware is installing. You need to use a freshly formatted (in camera) and empty memory card in the camera, during a firmware installation.
There aren't a lot of other things you can try yourself. The camera might need to be looked at by a technician, either at Canon service center or an independent repair shop (if the camera is out of warranty).
