Generally the term emissions or RF energy is used rather than radiation. While radiation or even non-ionizing radiation can be used it causes excessive worry and panicking in people who don't understand the differences between types of radiation. No one claims that a lightbulb is flooding the room with radiation but it is, we normally call this radiation light. And the light from a bulb is both more energy that a phone emits and a lot closer in frequency to the dangerous types of radiation than anything you'll get from a phone. In other words if a phone is bad for your health then you need to spend your life in a dark room because the sun or any artificial light will clearly kill you.
Things can be split into two groups - Intentional radiation and unintentional.
Intentional radiation is when the phone is using a radio transmitter. So any extra radio activity will generate more emissions.
When in standby the phone will exchange data with the network at a fairly low duty cycle, just an occasional I'm still here message. Anything that requires more communication will increase the radio activity. So any phone calls, texts, data etc... will increase the amount of information the phone needs to send and so increase the emissions.
There is also a hand off every time you move between cells which generates a small amount of data.
Independent of how busy the phones transmitter is the the phone adjusts its transmit power based on the signal strength so low signal strength will cause it to transmit at a far higher signal power.
So lots of data in an area with weak signal will put out a lot of energy (and drain your battery)
Other radios such as WiFi and bluetooth will also transmit however they are lower power. WiFi will reduce the amount of the phone needs to use the cell network for data and uses a lower power than the phone data link. All things being equal wifi can result in a net decrease in transmitted RF energy however people normally send more data when on wifi. Bluetooth is very low power, unless you are using something that requires constant data, e.g. using a bluetooth headset, the power it radiates is negligible in comparison to the other radios.
As with the cell phone radio, total emissions are very dependent on how busy the radio links are. These are battery powered systems, when not in use they are designed to be as low power as possible. Radio transmitters use a lot of power and so are kept powered down as much of the time as possible.
Finally there are unintended emissions, these are far far lower power than the intentional transmissions but can be measured. All electronic systems emit small amounts of RF energy, the busier they are the more they emit which means anything that uses the phones CPU is causing it to emit a small amount of RF radiation.
One minor addition: Connected cables can have an impact (headphones, chargers etc...). While they don't cause any extra emissions directly they can end up acting as antennas and so increasing the efficiency with which the already existing signals are transmitted. This is primarily going to impact the unintentional transmissions but it could also have minor impacts on the intentional transmissions.