Vibration and humming, slightly rough idle

She's got an automatic trans, and it definitely happens when the car is in park right at startup, without her foot on the throttle. I have been in there when it has also happened while backing up slowly (like out of a parking spot), but again, right after startup with a cold engine. I don't think she's ever tried giving the engine a little rev to see what happens.

The few times I've been in the car recently, what happens is you start the cold car and the whole car has a slight vibration (like a slightly rough idle), and there's some vibrating/humming noise from what sounds like interior trim in the rear. The noise lasts 2 or 3 seconds and goes away, but you can still feel the slightly rough idle. When you start driving, either the engine smooths out a little, or we just don't notice it anymore. It doesn't make the noise at stoplights, parking lots, etc, once the car has been running a little. The noise occurs both when the car is stationary and when it is moving, but only right at startup with a cold engine (so she usually has it stationary still).

I drove the car some this weekend and tried to feel the rough idle at stoplights and when parking, but it is not so rough that it stands out. Just like you don't have any doubt that the engine is running, kind of. But no noise, and I still think the roughest vibration is right after starting the cold engine.

Thanks guys. Hope I didn't forget anything else important.

Thanks for the information. My fiancee generally takes her car in for service (oil changes or specific problems) or has her dad do oil changes and simple work when she's home visiting. So she doesn't have a repair manual for her car. But we're 5 hours from her parents' house now, and we don't have money right now to throw away on shop repairs we could do just as easily ourselves. I like to do whatever work I can on my own car, but that generally has just meant brakes, spark plugs, oil changes, air filters, and other simple stuff. So before we take it in to a shop or dealership and end up paying for work we probably don't really need, we were hoping to get some information on here.

You have been very helpful with your suggestions. It's also great to see that one of the mods lives here in town. We're new to Tennessee, so that's especially helpful.

I think I'll go ahead and do the spark plugs and wires this week and see if that helps. As for the engine roll restrictor, I've read about that on here in my searches, but I couldn't think of a time she'd been through a lot of gravel and debris, and it sounded like that usually happens immediately before the problem becomes apparent. Still, she comes from a small town that probably has a lot of gravel roads and parking lots, so it sounds reasonable. I'll do a search for what that part looks like and where exactly it's located. If anyone can point me toward some pictures so I know what/where I'm looking for, that would be great.

Thanks again. I'm sure I'll have more questions for you in the future.