this sounds like it may be misfiring on 1 or more cylinders, which then cease misfiring as the engine warms
So I'd begin by replacing the spark plugs if any of them fail this checklist:
You can also try spraying some water on the coil pack with the engine running. It is possible the coil is faulty and causing the misfire. If you see sparks arcing with the water, then replace the coil.
I'd check the condition of the injector plugs / connections as this could be a lean misfire if an injector isn't firing properly - the best way to check this is with a 'noid' light
Then check all the connections to the engine sensors one at a time, remove each one and examine for signs of corrosion or damage and a squirt of contact cleaner ( not WD40) will also help.
This could be an airleak after the airflow meter, any air dragged in here isn't 'seen' by the ECU and so not compensated for and leans the engine out causing rough running.
Check the hose clips for tightness and inspect the trunking for any cracks or splits and also all the vacuum system, the small bore pipes and fittings for cracks and missing parts.
The best way to locate a leak is to have the engine running and warm and then spray lighter gas around each joint in turn. If the engine rev's up you've found your leak.
Work your way through each possible joint one at a time and you should find it. I use a slightly flattened piece of brake pipe and some rubber hose from the can of lighter gas to provide a spraying 'wand' and allow a direct blast of gas into each area, especially those difficult to reach with large implements.