Village boats to Manono depart from a landing just south of the Samoa Village Resorts at the west tip of Upolu. The Falelatai, Manono, and Samatau buses from Apia will bring you to the landing, and the boat leaves soon after the bus arrives.
The boat operators always try to charge tourists ten times more than the Samoan passengers for a special charter. They'll tell you the next "regular" boat won't be for four or five hours, but if you say that's fine and sit down to wait, they'll begin loading the Samoan passengers after about 10 minutes and you'll also be able to go for a small fee.
There are two landings on Manono, at Faleu on the south side of the island and at Salua on the northeast side. Ask the locals what time the last regular boat goes back to the mainland.
Some travel agencies in Apia offer full-day tours to Manono. Rather than taking the tour, get a small group together and charter a boat to Manono from the landing. Make very sure the price you agree upon before is clearly understood to be for the whole boat and not per person. Pack a picnic lunch, because little is available on the island.
You could also visit nearby Apolima Island, in the strait between Upolu and Savai'i, this way by chartering a boat. It takes about two hours to hike right around Apolima, but you're supposed to be invited before you go.