from English Grammar Today
We can use the verb suggest with a noun phrase, a that-clause, the -ingform of a verb or a wh-question word (where, what).
Suggest + noun phrase
We can use a noun phrase as the object of suggest:
Can you suggest a good restaurant in this part of town?
If we need to mention the person who receives the suggestion, we use a to-construction:
My teacher suggested an exam I could take at the end of the year. (or My teacher suggested an exam to me which I could take at the end of the year.)
My teacher suggested me an exam…
Suggest + that-clause
When we suggest an action to someone, we can use a that-clause. In informal situations, that is often left out:
I suggest (that) you leave here around four o’clock. The traffic gets very bad from about 4.30 onwards.
When suggest is in the past, we can use should in the that-clause:
Her doctor suggested that she should reduce her working hours and take more exercise.
Suggest + -ing form
We can use suggest with the -ing form of a verb when we mention an action but do not mention the person who will do it, because it is understood in the context:
She suggested travelling together for safety, since the area was so dangerous.
Suggest + wh-question word
We can use suggest with a question word such as where, what, who, how:
Could you suggest where I might be able to buy a nice sweater for Mark?
Can anyone suggest how we might attract more people to the next event?
We don’t use suggest + indirect object + to-infinitive when we suggest an action to someone:
He suggested that I should apply for a job in a bank or insurance company.
He suggested me to apply…
We don’t use suggest + to-infinitive:
Henry wanted to tell everyone, but Dora suggested waiting until the news had been announced officially.
Dora suggested to wait…
(“Suggest” from English Grammar Today © Cambridge University Press.)