1996 Chevy Corsica engine slow to turn over. Once started

Just bought a 96 Corsica 3.1 Starting problems have occurred since the first few days, sometimes more than others. I live in Minnesota, and it was dang cold here for the first few weeks I owned it, so I let the problem go, since, eventually, by holding the gas pedal down a little or after driving a mile or so, the car seems to drive fine. Also, the engine has never fully died, unless I didn't give it any gas when cranking the ignition. I don't have a tachometer on the dash, so I can't say for sure about the car's idle RPMs, but once I turn the engine over with a few small pumps of the pedal, it sounds to me like the rpms go up and down a little. I looked under the hood and the entire engine is jostling back and forth a bit. I shift into reverse or drive and sometimes the idle speed is high-the car wants to go 15-20mph w/o any push on the gas pedal. Any thoughts on what this problem might be? To be thorough in my information I am giving, I will include the info that a mechanic told me before purchasing the vehicle. There is seepage of coolant in the lower intake manifold gasket. The tranny fluid was dirty. I have done nothing to correct these two problems yet. Fluids are staying level. The tranny fluid appears cleaner now (weird). I had an oil change done, I bought a new air filter, and I used some Chevron fuel injector cleaner gasoline additive to a tank of gas, and another brand to another tank. These additives increased my gas mileage from low 20s to near 40mpg, but only for the tank of gas I add it to. My next thought was to replace the fuel filter. Thanks for your time. I am very open to suggestions. I am on a pretty tight budget for a month or two; hope I resolve or delay expensive repairs. Thanks again, Joel