he suggested visiting Jane: this means that he, in the past, suggested to someone/s that they could visit Jane.
He suggested that they should visit Jane: this means the same, but the suggestion is for the future, that would be the present. That is to say, He suggested that they should visit Jane NOW. In the previous one, the suggestion was still for the past. The word "should" also gives a sense of obligation.
He suggested that they visit Jane: This means that the same as the first one.
He suggested that they visited Jane: This is the same as the second one, but without the sense of obligation.