The most successful method is a mix of both.
Start with only or mostly old and cheap aircraft, let's say 80% used and 20% new.. Then there's a small time frame where you have old ones and new ones at the same time, about 50% old ones and 50% new ones. This is where you should at last stop ordering old ones, except to exchange a very old one (18 years or D-Check soon) with a better short lease of the same type (7 years old, D-Check in 3 years for example).
If you miss the correct time frame to replace mostly or all of the old ones, you will fail.
If you replace them too fast, you lose money and leave the path of effectiveness.
At the end you should never hesitate to do strict changes if the game hits you hard. If fuel prices rise extremely and your old aircraft stop making money, give them back or minimize the losses by grounding them and retire the crews.
A pure new aircraft start has some advantages (sometimes same type for the whole game, enough production slots, a new and fresh looking airline, no anger because of the used market situation) but it is never the most effective way and in same cases it can destroy your airline, mostly if you are at a very slot restricted airport.