Turn the key from off/unlock to on (not starter engage), to off, to on, to off, to on and leave it there, then look at the odometer numbers and the codes should pop up. You can look the numbers up from there, and/or post them here and we can help.
The 'ck eng' light is separate from any oil change reminder. They are not related.
The yellow 'ck eng' icon in the cluster indicates that there is a fault code stored in the PCM.
Any oil change reminder would be in the overhead console display (EVIC), if equipped.
On the van you have to push the trip or reset button to get it to display the codes. I forget which and I don't have my 99 van any more. But it displays the old two digit codes and will probably have a code 23 which you should ignore.
The best bet is to find someplace like AutoZone that will read the codes for free. You'll get the newer OBDII codes that way.
You won't start the engine. Actually the procedure in the video will only report errors with the instrument cluster. The one for the trouble codes is similar. Try this procedure
Dan Stewart wrote about the 1999+ Voyager/Caravan:
Get Diagnostic Trouble Codes by placing the key into the ignition. Push and hold the both the "Trip" and the "Reset" buttons with the left hand and turn the key to the "ACC" position (first position between "LOCK" and "RUN"). Continue to hold both buttons in and count to 5. Release both buttons, display should read trip mileage instead of total mileage. Push the "TRIP" button and the odometer will display DTC'S in rotating sequence instead of displaying total mileage.