The best way to “dispose” of cellphone is to recycle it with your carrier or through an electronics recycling program. Cellphones tend to be very easily reused. Most the US National Carriers (and I would imagine most carriers world wide) will take back phones when you upgrade. The phones they take back are assessed and then redistributed to a number of different channels: free phones for domestic violence victims, phones for elderly folks to call emergency services, phones for specific uses such as national parks staff and finally phones which get refurbished and resold.
There is a very large secondary market for all types of phones. Sometimes the phones are sent to the ‘insurance’ companies and refurbished to be redistributed as replacement phones. Other companies focus on selling high quality used phones via stores (online and physical) and other companies demanufacture the phones into viable components that can be resold to help refurbish other phones or used in other electronic device manufacturing.
If your carrier doesn’t accept the phone for recycling, in the US larger retailers like Walmart, Target and Best Buy will take back the phones in their stores. These phones are then sent to a 3rd party company to manage one of the above channels of repair/reuse. Similarly if your community has or offers an electronics recycling program locally, cellphones are typically accepted at these events as well.
You should not put your cellphone into a trash container or your curbside recycling bin/cart. Cellphones contain rechargeable batteries (as well as valuable components) and batteries should never be mixed with paper. Rechargeable batteries could wind up causing a fire if they are damaged, and mixing them with paper destined for a recycling program could wind up being a very bad event if it catches on fire. Similarly, rechargeable batteries contain heavy metals and other toxic materials that should not be placed into a landfill. Proper management of your cellphone is definitely a great thing for the environment!