The sentence "I will do my best to please you" is correct. "To please" is a transitive verb meaning to make (someone) happy. For example, "The flowers pleased me" means "The flowers made me happy." The word isn't very common in speech, so I wouldn't really recommend using it. But in this case, it means, "I will do my best to make you happy."
Now.... "suppose," "guess," and "suspect." It's very to explain the differences between these words. They ARE very similar, but there are slight differences in connotation. I will try my best!
All three words show a mix of belief and doubt. Sometimes someone says something, and you're not sure if you believe it, but you don't want to argue. It's common in that situation to say "I guess" or "I suppose." (Or "I guess so" or "I suppose so.") You do not use "suspect" in this situation.
You can also use "I guess" and "I suppose" to introduce statements you're not sure of. In my experience, we usually use them when talking about what other people say or mean. For example, "I suppose he's right" or "I guess he knows what he's talking about." It's less common, but permissible, to use them in other situations. "Do you suppose the store is open this late?"
"Suspect" is a little different. It often has a negative connotation. ("I suspect he's lying!") It is sometimes used to introduce a new, even subversive statement. That is, you can only "suspect" something if it isn't common knowledge. "I suspect" often implies this: "Other people may not believe it, but I have a feeling that it's true."
In most cases, though, I think you would be fine to use any of the three. If you have any specific questions about when to use them, please let me know!