Completely natural and correct.

Tim's sentences are all good too, but I agree with Chris -- your original sentence is perfectly natural. You don't need to change anything.

Yes that sounds like something a native speaker would say. It would sound better if that wasn’t the end of the sentence though because your sentence sounds a little like an unfinished thought. Not because of the grammar. Something like “He suggested going fishing tomorrow.” sounds more natural. Another option is to specify who he suggested go fishing so there doesn’t need to be two gerunds. For example “He suggested they go fishing.”