We highly recommend flying into Monterey Regional Airport (MRY) as it is the most convenient. For those flying from Denver, there is a direct flight on United from DIA to MRY, but it can be more expensive. Other options also include flying into San Jose International Airport (1hr 10min drive from Monterey) or San Francisco International Airport (about 2hr drive from Monterey). There are available shuttle reservations that can bring you to Monterey/Carmel from either SJC or SFO. A reliable shuttle service is Groome Transportation (formerly Monterey Airbus Shuttle).

Renting a car would make traveling a lot easier, especially if you are trying to do some sightseeing in the Carmel/Monterey area.

Yes, but they are sparse in Carmel. In Monterey, there are more available. Both Lyft and Uber.