If you've got some random PIR hardware that you don't have a datasheet for, you get to reverse engineer the interface. It may be as simple as a pin that goes HIGH when motion is detected, or you might spend the next month decoding a proprietary protocol using an oscilliscope or logic analyzer. Probably easier to buy a new one that comes with documentation.
The doorbell press should be easy, get a relay with a coil that matches your doorbell transformer (typically 16VAC in the US), then connect an Arduino digital pin in INPUT_PULLUP mode and one of the Arduino's GND pins to the NO relay contacts.
As for the weather stuff, look into Dallas One-Wire devices. They're popular and there's even an Arduino library for those.