If you can car spot the Androscoggin River has a lot of nice spots. There are rentals at White Birches Campground in Shelburne. Bethel Outdoors in Bethel also rents and shuttles. A nice place is Reflection pond in Shelburne, if you put in at Rt 2 you will need to portage over the RR tracks. Shelburne to Gilead is a nice stretch, you do need to check out the take out point in advance as its easy to miss. The stretch from Shelburne Dam is quite nice with lots of islands but the water is slightly rougher than putting in at the Dam.
If you want more challenging water, there is a carry in boat launch at the intersection of the Peabody River and the Androscoggin behind the town garage. You can drive back to the PSNH power house and carry the boats down the long gravel path between the rivers. The stretch to Shelburne is a lot wilder than you think, lots of wildlife and islands but it can get shallow.
The rafting outfit in Gorham may offer rentals. A general note is if a place has a rental spot for boats its probably not going to be quiet.
Jericho Lake and South Pond along Rt 110 are both nice spots with little no traffic. South Pond may get some power boat traffic from the developed end, while I beleive Jericho is no power boats (the trade off is lots of ATV traffic in the area) You will need to rent down in Shelburne although I think the Milan general Store may also rent. The paddle from the Milan bridge to Berlin is nice. I cant recomend Cedar Pond or Nay Pond as they get a lot of boat traffic. The Pontook dam area is flat water, lot of moose, osprey and eagles, no need to car spot and planty of paddling but there are some powered fishing boats.
If you want to go with a group, I can recommend these events for the next two weekends
http://arwc.camp7.org/. Unfortunately you missed the local events last weekend.