How to pick the airline to fly to Bali ?

Airline tickets vary in prices and conditions and more than
one airline can bring you to Bali.

The big difference between the airlines are in the route they take and from whom you will purchase the ticket from.  

Since routes are changing quite often, it makes sense to ask an agent about your particular flight, chances are they know or can find out. Experiences tell it's still 50% uncertainty left after they advice you, because schedules change faster than the normal travel agent can keep up with.

Many times the cheapest tickets involve the most stop overs en route, so it actually makes sense to pay a bit more to have a shorter flight. My personal experiences only cover exit airports in Europe like Amsterdam, Frankfurt and London. Here there are at times huge differences among the airlines.

Thai Airways had always those lengthy 4 hour transits in
Bangkok that made the flight to Bali about 18 hours long,
Garuda stopped a lot at different airports to make sure the
plane was always packed full - a 21 hours trip, ouch!
Really convincing was only Singapore Airlines, nonstop to
Singapore, after 45 minutes transit straight to Bali. 14 hours after leaving Amsterdam we finally reached Balinese soil.

Interesting can also be Quantas flights going to Australia
that have a stop in Bali. As I said it depends where you fly
from and how much you are prepared to pay for the ticket. Best is always a nonstop flight, nonstop Singapore is fine too. Be aware that a "direct" flight is not necessarily also a nonstop flight.