Here are some phrasal verbs with 'on'. Let's go on. Let's carry on. Let's press on. (If the task is long and boring) Let's push on. (Ditto, especially for a journey).

It depends on the context, but you can usually say: "Let's keep going." Let's carry on." Sometimes, usually when you are talking about something serious and enduring (lasting a long time) like the death of a friend, you might say: Let's (try to) move forward."