The key job of a roof is to keep water out. But it should also be energy efficient, and that means that the roof should also keep the heat out of our homes in hot weather. Yes: keep the heat out… even in cold climates.
Roofs can’t ever be a significant source of useful heat gains.
Attics have to be properly insulated to prevent heat loss during the heating season - and that and the need of protecting the house from the summer’s sun, excludes the possibility of heat gains during the heating season.
Roofs & Roofing Basics 
Avoid complicated rooflines: they are more prone to leaks and ice accumulation. And are more expensive. Simple hipped and gabble roofs are the best choices.
Avoid details like dormers (they may look charming, but are difficult to seal and flash), or chimneys (new homes do not need chimneys) and skylights, as much as possible.
There isn’t a “best” type of roofing. It depends on local climates, prices and practices. Preferences count.
Metal roofing - steel roofing and the more expensive standing-seam metal roofing - is a good and flexible choice, in many climates. It lasts longer than asphalt shingles, and can be recyclable.
Concrete tiles are fragile; cedar shingles are beautiful but flammable. Clay ceramic tiles and asphalt shingles can be a good choice in countries and regions where they are common and cheap.
Whatever the type of roofing, pay special attention to installation details involving the underlayment, step flashing, plumbing vent pipes and every possible roof penetration...
Attic (roof) insulation is critical to avoid overheating in hot weather, and to avoid heat loss to the attic in the winter. The materials that make the ceiling and roof do not prevent heat transfer. Only insulation can prevent it. The comfort of your new house will largely dependent on attic insulation. Obviously you also need a very tight attic.
Avoid flat roofs and the cathedral ceiling design, if possible. They can be a source of problems.
Attic InsulationAttic insulation is the most critical factor to reduce heat loss in winter, and solar heat gains in hot weather.
Consider very high levels of attic insulation - and a very well sealed attic….
See:
Attic insulation
Overhangs should be a key part of your new home's design.
Overhangs are critical for keeping rain water away from your walls and the foundation of your house, and to protect the windows and doors from rain water (or the sun, in hot weather).
See: Roof overhangs for summer shade and moisture control
Flat roofs are difficult to insulate and to seal properly; they can be a source of problems. Similarly, cathedral ceilings are a source of air leaks, rot, condensation, ice dams, energy waste.
Avoid them, in new construction - except with very good reasons.
More:
Cathedral and flat roof insulation
Radiant Barriers Are Easy to Apply but are not Effective in most climatesReflective radiant barriers - sheets of aluminum or other high reflective material applied to cardboard, kraft paper, cardboard, plywood, or other substrate - are lightweight, clean, non-toxic and easy to handle and to install.
But they may not be effective, except in hot climates, in houses with low levels of attic insulation. Don't be fooled by manufacturers' claims.
See: Roof Insulation
Roof Ventilation Can be Advantageous in Many cases, But Attic Ventilation Fans May Be IneffectiveRoof ventilation can help keep attic temperatures lower. Just do not overestimate its importance, and do not increase ventilation beyond the necessary.
And be cautious with attic ventilation fans; they can be very ineffective.
See:
Increasing Roof Ventilation
Roof Ventilation & Energy Efficiency
Attic Fans Efficiency
Reflective Roofing MaterialsLight-colored roofs are great for roof reflectivity and to lower the attic temperature. But reflectivity depends a lot on the type of roofing.
See:
Metal roofs Energy Performance
Shingles vs. Clay Tiles Energy Performance
Membranes for Flat RoofsPoly-iso and other spray foams are excellent choices for flat roofs. They are very durable and do not leak. But their yellow color is not attractive.
Plastic and rubberized membranes like EPDM (Ethylene Propylene Diene Monomer) are more popular, and are energy-efficient choices. White EPDM rubber membranes provide very high levels of reflectivity (80%).
See: Energy Efficient Roofing Membranes
Roofing Coatings Are Inexpensive but May Not Be Compatible with Existing Roofing MaterialsReflective roofing coatings can prevent overheating, in existing roofs. Materials like acrylic elastomeric coatings are easy to install on many existing roofing materials, and will extend their life.
The problem with them is that they are not compatible or effective with asphalt shingles and other common types of roofing materials.
See: Energy Efficient Roof Coatings
Eco-Friendly roofingAsphalt shingles rarely last more than 20 years; and that’s too short...
If possible, choose a more durable roofing material. And consider eco-friendly features.
See: Green and Energy Efficient Roofing Materials
ChimneysNew appliances can draw air for combustion from outside the house directly onto their burners; and can vent exhaust gases to the outside, directly, with obvious advantages: there isn’t air conditioning loss or air infiltration. It’s safer and can save a lot of energy.
In this sense chimneys are largely useless in new buildings, except perhaps for those living in remote areas, where wood heating makes sense.
See: Do new homes need a chimney?
Green RoofsThere is much talk lately about green roofs.
Why not create “gardens” on the top of our buildings? But are they practical and advantageous for the common homeowner?
See:
Green Roofs Pros and Cons
Types of Green Roofs
Read our book (Print and Kindle ebook versions);
Save thousands of dollars;
Avoid costly mistakes and pitfalls.
Follow us: 
©2013 House-Energy, Help Homeowners reduce energy use and carbon emissions. Ed. Reisinh