Believe it or not, I have ridden this exact route albeit a few miles short of New Quay.
http://ridewithgps.com/routes/4453723
Hills are unavoidable, with the worst sections coming within the last third of the route. However, it is a lovely fell route.
Between Shobdon and New Radnor, you can go either via Kington or Presteigne. The link shows Kington but Presteigne is a much quieter route for traffic and preferable.
Beyond Builth, avoid the tedium of the A483 by taking the slightly hillier parallel lane on the opposite side of the valley. After Beulah, you enter a gorgeous valley before turning right towards The Devil's Staircase, after which the climbs become savage for approximately 10 miles, and there are three of them: The Staircase (short but very sharp and best to push) the long drag up The Gamallt and Telephone Box Hill. If you are loaded with luggage then this section will be challenging. See the following Youtube clip of the Milk Race:
https://www.youtube.com/watch?v=y-qCNTlFTGM
Tregaron to New Quay is not easy. After Felinfach, there is a pretty stiff long ascent before the road sweeps down to the coast.
Cafes:
Tenbury
Shobdon Airfield (just off the route)
Kington or Presteigne
Builth Wells
The forestry cafe
http://www.visitmidwales.co.uk/Llanwrty ... ue=1013903
Tregaron (although I have never found either of the two to be open). There is a Coop
Rhayader (on the northerly route)
Alternatives:
The southerly approach is via Llandovery and Lampeter but considerably longer. Although you don't reach the same amount of ascent as going via the Staircase, this diversion is far from flat and it uses a moderately busy and boring A road. If forced to use an alternative to the Staircase, the only other option is the northerly approach.
The northerly approach can be made via Rhayader and the Elan Mountain Road. This route is rewarding in terms of scenery but still requires climbing. The ascents are less severe but they are very long and Alpine in style. You will still have to suffer the hills inland of New Quay.
Don't use the coastal A road, it is unpleasant and congested with traffic. It is not flat either!
The other alternative is to catch a train to Builth and cycle the remainder.