check for a vaccum leak start the engine and spray some starting fluid around the gaskets and fittings on the intake manifold if the engine speeds up there is a leak
replace timing chain
Starts hard then it rough idles for a short time then smooths out but when you give it gas it miss fires and bucks and does not smooth out.
Sounds Like You Are Out Of Time. Most Likely Timing Chain.
sounds like the timing is off
Motor mounts are bad
Mine did this and I found out it was the timing belt that needed to be changed.
check your timing & up your idle a little
Rebuild it. If it's a 97, it must have close to 150k miles.
Check timing. idle could also be set too low (rpms) so vibration is increased.
Timing belt could be bad and could have jumped a tooth or you need a toon up.
I have 1998 a Lincoln town car with the same problem.. It idles rough when you start, then smooths out once it warms up and you give it gas. From what I am reading its either an oxygen sensor or you have vacume leak in one of your hoses by the manifold.
I'm not a pro but sometimes the timing can be off, if it idles higher than it should out the gate, that can do it. That's all I got...good luck