Suggest is one of those verbs that cannot be followed by a to-infinitive.

Suggest can be followed by an –ing form or a that-clause.

• Activists suggested abolishing the plan to build the bridge.
• OR
• Activists suggested that the plan to build the bridge should be abolished. (NOT Activists suggested to abolish the plan to build the bridge.)

• The doctor suggested that he should quit smoking. (NOT The doctor suggested him to quit smoking.)

Suggest cannot be followed by a direct personal object. You should use a structure with to.

• Can you suggest a nice restaurant to me? (NOT Can you suggest me a nice restaurant?)
• I don’t know what I should suggest to her? (NOT I don’t know what I should suggest her?)
• He suggested to me that I should change my style of parenting. (NOT He suggested me that I should change my style of parenting.)

Suggest is one of those verbs that can be followed by a subjunctive structure.

• He suggested that I write to her as soon as possible.
• She suggested that I accompany her.

In British English, these structures are not very common.

In that-clauses, British people usually prefer should + infinitive.

• He suggested that I should write to her as soon as possible.
• She suggested that I should accompany her.