Start with a TB clean, usual diagnostic for polos
Missfires, could be anything on the side of ignition at a guess, faulty sparks, HT cables, distributor? Another possibility temp or lambda sensor. I'd start at the plugs & ignition system, possibly start with a spark plug replacement, if no luck look at other options or an electronic diagnosis. Check also when it was last serviced if you can as that might lead you to suspect it to be a service based fualt or not.
Maybe a wrong firing sequence...? (doubt it as it runs fine once warm). Other suggestions to the problem are:
Haynes says for backfiring:
Timing belt incorrectly fitted or tensioned belt
Vacuum leak at TB or manifolds
Blocked Fuel injectors
Ignition Timing incorrect
As for engine difficulty start when cold from Haynes:
Weak/discharged battery (I had this, battery was really unreliable)
Ignition timing incorrect
Warn spark plugs (very possible, as they crack on the inside, so once they heat up the contact is remade with the broken filament or so I've heard)
Fuel injection fault
Ignitions system fault
Low cylinder compressions
Hope this helps.
