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.