1999 Ford Taurus • 156,000 miles

I just started having this issue today with this car, it's a POS really but I have a four year old and a hubby with a job so I need my car. Anyhow the issue is when I accelerate at like a stop light or something I can hear and feel a thump noise from under the car, either in the front or back. I'm not really sure what this issue is. I know the car needs some repairs, the main one that I know of is a coolant leak which I've been having to fill the coolant every now and then. Any help with figuring this out would be nice, I have a 1999 Ford Taurus.
December 27, 2012.

Is this a one time "thump" when you acellerate from a stop or is it a "thump, " "thump, " "thump" continuing as you acellerate until a certain speed is reached?

Single "thump" sounds like a bad motor mount or other loose engine or suspension part.

Dec 27, 2012.
It's usually just one thump, there's been one time where it was multiples but it wasn't right after another and it was happening when I was in a 25 zone. My brother in law is saying it may be the transmission dying and I'm really hoping that's not the case.