This video shows you how to get to Wat Pho and the Grand Palace by taking the Bangkok MRT. You should take the MRT Blue Line to Sanam Chai and leave via Exit 1. The Wat Pho, also known as the Temple of the Reclining Buddha, is around a 5 minute walk away while the Grand Palace is further down the road.
I stayed quite close to the Si Lom Station which is on the Blue Line. I entered the station and bought a ticket using one of the machines. I tapped my destination on the touch screen and was told to pay 26 Baht. Afterwards, the ticket popped out from the slot at the bottom of the machine in form of a small blue token.
I went through a security check before entering via the ticket gate. I simply had to tap my token on the reader on a ticket gate to open it. You may also use your debit or credit card if it can be used to make contactless payments.
I took an escalator down to the platform, boarded the train and reached Sanam Chai in 9 minutes. In conclusion, the MRT is a fast and cheap way to travel around Bangkok and a good way to avoid all those traffic jams which are very common in the city. Taking the MRT allows you to avoid getting overcharged by unscrupulous taxi drivers and saves you from having to wait 10 minutes or more if you try to book a ride using the Grab app.