You can use the subjunctive after these verbs:

Propose            recommend       insist     demand

• I suggest (that) you take a vacation.

• They insisted (that) we have dinner with them.

• I insisted (that) he have dinner with me.

• He demanded (that) she apologize to him.

You can use the subjunctive for the present, past, or future:

I insist you come with us.

They insist I go with them.

Other structures are possible after insist and suggest:

• They insisted on my having dinner with them.

• She suggested that he buy some new clothes. (not suggested him to buy)

• What do you suggest I do? (not suggest me to do)

Should is sometimes used instead of the subjunctive.

• She suggested that he should buy some new clothes.

• The doctor recommended that I should rest for a few days.