You should + (verb)

Here you are suggesting an obligation or duty that needs to take place either now or in the near future.

Here are some examples:

“You should go to bed.”
“You should do your homework before going outside.”
“You should replace your headlights on your car.”
“You should request a raise at work.”
“You should stop smoking.”
“You should smile more.”
“You should slow down when driving in a neighborhood.”
“You should talk to him about it.”
“You should train your dog.”
“You should trust what they say.”

You’re supposed to + (verb)

‘You’re’ is a contraction of the words ‘you are.’ When using ‘You’re’ with the words ‘supposed to’ you are making a suggestion that something you strongly believe ought to happen.

Here are some examples:

“You’re supposed to keep that secret.”
“You’re supposed to let me know when you leave.”
“You’re supposed to stop when at a red light.”
“You’re supposed to unpack once you get there.”
“You’re supposed to return the movies you rent on time.”
“You are supposed to remain calm.”
“You are supposed to fasten your seat belt.”
“You are supposed to invite all your friends.”
“You are supposed to encourage one another.”
“You are supposed to decide before next Thursday.”