Car runs rough first mile after sitting overnight

May just be dirty gas...how long since tune up? Try 1-2 bottles of Lucas Oil Injector cleaner added to gas tank, do you know how to clean the throttle body...check for clogged/dirty air filter too...Hope this helps.

Can't hurt.
I would change the spark plugs first.
No check engine light when you are driving ?

check if it has a crankshaft position sensor if it has test it-

Use a high pressure AIR jet so that the water is blown off and also use a hot air dryer on the coil.
It is possible that the wet coil has reduced the firing intensity and made the engine rough with a change of timing.

Not why the check light came on ,so no.

Get it diagnosed,they use the codes to
find a solution to your vehicles issues and
repair.

My rule of thumb for Timing belts:
replace at 60,000 miles, and every 40,000 miles thereafter. Having said that, I often let
cars go until the timing belt starts showing signs of wear (usually every 50K - 75K miles)

But www.autozone.com (register free at www.autozone.com) is recommending for your
car driving in "severe driving conditions" (aka normal stop-start driving) that your 2000 Sonata
have:
60,000 miles - replace
90,000 miles - inspect for wear
120,000 miles - replace
150,000 miles - inspect for wear

The 155 page owners manual (I found one at the link below - you may prefer to try Hyundai)
makes no mention of the timing belt at all.

http://www.scribd.com/doc/29113477/2000-Hyundai-Sonata-Owners-Manual

For the 2005 Hyundai Sonata
I do not know if you have the 2.4L In-Line 4-Cylinder or 2.7L V6; however I have included both below....

For the 2.4L In-Line 4-Cylinder:
Firing Order:
1-3-4-2
Cylinders:
..........(back).........
|...1....2....3....4...|
.........(front)

For the 2.7L V6:
Firing Order:
1-2-3-4-5-6
Cylinders:
.......(back)........
|...1....3....5...|
.......................
|...2....4....6...|
.......(front)
Coil Connections:
(back)
...3
...6
...2
...5
...4
...1
(front)

Let me know if this helped, or if you have additional information or questions. Feel Free to contact me at FixYa.com!

Try some dry gas, there maybe water in the tank or crud being kicked up,

Next time fill it up when it gets to a 1/2 tank to test..

if the problem remains it could be a evap or pressure problem..

How old is the fuel filter?

Check your fuse and relay to the radiator fan, There going to be high amp fuse so most likely will be in the fuse box by the battery. If the fuse and relay is good, check the wire harness connection at the fans. The other thing to is to run the engine and turn on your A/C, the A/C should over ride the switch and turn on the fans. If the fans do turn on, inspect the radiator fluid, make sure it's at it's level and if it's never been serviced. It may be time, Orange antifreeze has a life of 150,000 miles or 5 years and the green antifreeze has a life of 30,000 miles and 1 year. The thermostat may also gone bad but i don't think at this case since your car only heats up while your stopped. If everything is working (fan) just replace the antifreeze 50/50 mix (Antifreeze/water), Water has great heat transfer but boils so antifreeze is added, and 100% antifreeze does not have great heat transfer and will gel when heated so the two have to be mixed 50/50 and if your not sure how to mix the two. You can buy them 50/50 mix and the capacity to your car total fill is 7.3 liter 4 cylinder/ 8.6 liter 6 cylinder. 

Sounds like its either fuel or ignition. Have someone turn the car over while you sniff the exhaust.
If it smells of raw unburnt gas , fuel is not the issue , your problem is most likely ingition. Car might run rough at first when it does start , due to being slightly flooded.
If no gas smell probably a fuel problem. Possible low fuel pressure , or no fuel injection.