Hello Dinghui! As an English learner, I am interested in your question either, so I googled it and I suggest that you take a look at this detailed explanation http://www.phrasemix.com/answers/how-do-you-use-suggest-correctly

Not exactly, but you could say "I suggested that he wash the dishes." or "I advised him to wash the dishes."

It's not a really an expression, but a common structure. The above link from Natalya is good for learning how we use it. I would add as a minor point that "suggest" is a verb that uses the subjunctive in English. In Kole's answer "I suggested that he wash the dishes." Usually we say "he washes" but this is the subjunctive, so it becomes "he wash"

Normally we would "make a suggestion to someone" or "offer someone a suggestion" rather than "suggest to someone". The best structure is: I suggest that you do _____. (this "that" clause must be in subjunctive mood).