My understanding is:
Pressure is the fact that force is being put on something, regardless of the effect on the item that receives the pressure
Stress is the effect on the item itself, it has stress due to the pressure that is exerted on it.
Strain is the effect of the stress, when the stress is too great and it is causing damage.
Eg. The manager put pressure on the committee to finish the project by February. The committee members were stressed. The stress caused a great strain in the company, as some of the employees ended up quitting, there was a shortage of workers, and the owners ended up having to fire the manager and hire a new one.