Some useful info in this thread.
And in that thread I gave the following advice provided by a roofer friend. It applies primary to shingles. (He gave it to me 20 years ago, so some of it might be dated.)
Insist on 30 lb. felt paper, not 15 lb. It is a little more expensive but well worth it.
When the crew leaves for the day, no felt paper should be exposed, as high winds can tear the felt. This is called “weathering it in.” This usually requires the roof to be done in sections.
Make sure they use plastic caps when laying felt paper. They should not use use regular roofing nails.
The roof should be shingled in sections (i.e. tear off, felt paper, and roof on the same day.)
They must first put a starter row down along the gutter. But before that, they should put metal down (drip edge). On the sides, they should install rake edge. The metal goes on-top the felt paper, then the starter row, then the stair-step. For the drip/rake edges use brown baked on enamel finish, not silver finish.
Have them put “ice and water guard” down on low sloped pitch areas. This must be bonded to the wood directly, so all felt paper must come off. Make sure they don’t nail it down – it must be stuck down. Felt paper is not needed. Inspect wood here, too.
On the chimney, if they replace the flashing, make sure they use ice and water guard around and up the chimney, and then put metal flashing on top of that.
Make sure they use roof-over-ridge vent. When they perform the cutting, make sure its not so deep that they cut the trusses, and make sure it is 2" wide at a minimum. It should be cut evenly on both sides (important).
Install new pipe collars on all drain vents.
Get a copy of their liability insurance. Get a copy of their certificate of workers comp. Get and follow up on 4 recent references.
Use 25 or 30 year shingles from a reputable company (e.g. Certainteed, Tamko Heritage, GAF-Elk).
Pay nothing up front, and 100% upon completion and my satisfaction. Get a minimum of a 5 year no leak warranty on labor.
Sign contract in the beginning. Everything should be in the contract, including picking up debris and hauling it away, felt paper, color, type, warranty, etc. Include all of the above stuff I mentioned! Contract should also have a start data and completion date. The contract should also state, “The completion date can be altered based on weather, but work should be completed ASAP in a prompt and professional manner.”