There may be exceptions (It's getting late and I'm pretty tired) Suppose usually means one person is giving a hypothetical situation to produce a different outcome to a question. - "You were really late for work, you missed the meeting." - "Suppose I had a second alarm clock, maybe I would have woken up then." Suggest usually means someone is directly giving advice to another person - "Wow, I was really late for work and missed the meeting." - "I suggest you get a second alarm clock, I'm never late anymore." If you have any questions feel free to ask.

"Suppose" is to make an assumption. "Suggest" is to offer a solution.

I suppose you are a learner of english (It means that I'm not realy sure but I think you are) I suggest you to listen to Luke's english podcasts (google it please) (it means I think it would be a good idea for you to improve your english)