Suggestions are usually reported using the reporting verbs suggest, insist, recommend, demand, request, and propose followed by a "that" clause. 'That' and 'should' are optional in these clauses. Note that suggest, recommend, and propose may also be followed by a gerund in order to eliminate the indirect object and make the suggestion more polite.

Examples:

A: 'Let’s go to the cinema' / 'Why don't we go to the cinema' / 'it would be a good idea to go to the cinema / 'I think we should go to the cinema'

B: He suggested going to the cinema.

B: He suggested that we should go to the cinema.

B: He suggested we should go to the cinema.

B: He suggested we go to the cinema

B: He suggested that we go to the cinema.

A: "I don't think it's a good idea to go to the cinema today. It's so sunny, we could go somewhere outdoors instead.'

B: He suggested postponing the trip to the cinema.

A: You should go to the British museum while you are in London.
B: She recommended visiting the British museum while I was in London.

In this last example, the tense of the verb 'to be' changes because I am not in London anymore when I report this statement. However, note that it may not always be necessary that the tense of the verbs in the reported clause changes.

Reported Speech (III): Reporting Verbs