The expression you are thinking of is "had better." There is a posting about the use of "had better" below on this Newsgroup, dated February 8.
"Would better" does not exist. Perhaps you are thinking of "I would..." in the contraction "I'd...." Confusingly in English, the contraction with "-d" can represent either "had" or "would"; you have to get its meaning from its context.
Here are some examples of "-d" meaning "would":
"¢ I'd rather be sailing = I WOULD rather be sailing.
"¢ I'd like another cup of coffee, please = I WOULD like another cup of coffee, please.
"¢ He'd be here by now if he had left on time = HE WOULD be here by now if he had left on time.
Here are some examples of "-d" meaning "had":
"¢ I'd already heard the news before Sarah told me = I HAD already heard the news before Sarah told me.
"¢ By the time I arrived, they'd already eaten dinner = By the time I arrived, they HAD already eaten dinner.
"¢ If you want to pass this course, YOU'D BETTER study every night = If you want to pass this course, you HAD BETTER study every night.
"¢ Since I'll be out of town next week, I'D BETTER finish this work ahead of time = Since I'll be out of town next week, I HAD BETTER finish this work ahead of time.
You are correct: "had better" is a strong suggestion, as in, "You'd better speak more softly." To express that idea in a gentler way, you could say:
You might speak more softly.
You could speak more softly.
Why don't you speak a little more softly ?
Is it possible for you to speak a little more softly?
Maybe you should try speaking a little more softly.