What should I do with the problem on my 2004 Ford Taurus

I have a 2004 Ford Taurus with 135,000 miles on it. I had the coil pack, spark plugs and wires replaced per the advice of the mechanic when the engine light came on. The code was P0306. After numerous times taking it in the engine light is illuminated (not blinking). It seems to be running ok but it is a little difficult to start. Instead of turning right over it may take 2 to 3 seconds especially when it is on a cold engine. It does not seem to happen on a hot engine. The diagnosis says it is cylinder 6 on the first 1,000 rotations. The mechanic that fixed it said I should take it to a Ford Dealer as they have better diagnostic equipment to pinpoint the problem. The Ford dealer wants $50 to $100 just to diagnose the problem. What should I do? Just keep going and hope it is a faulty engine light>

You have a misfire. You need to fix that before it ruins your catalytic converter.
It may be the number 6 fuel injector, or it may be something more serious. If you pay for diagnostics, make sure they verify the compression before throwing another part (the fuel injector) at it.

I second Tardis on all points.

Give up on the car. From experience it’s a money pit.

Ignore kizwiki.

The 2004 Taurus was actually a pretty reliable car. Your problem is likely easy and not even that expensive to fix, if a proper diagnostic procedure is followed.

Fine. Take the thing to any number of experts and spend a ton of money on diagnostics before you ever replace a part on an engine with 135,000 miles on it.
This is not a long life car. There are any number of problems waiting to unfold.
I had this car. It is good for as much as 150,000 or so miles. That’s pretty good for the money. But move on now.