Smoke and bad odor on startup

Hi, I am new to the forum so if I miss something please let me know. I recently purchased a 1993 chevy s-10 from my father at a price that I couldn’t pass up. The truck seems to run just fine, but on startup, a cloud of smoke comes out, whitish - grayish, not blue, and there is a burning smell. However, the burning smell doesn’t smell like oil burning and as far as I can tell I am not losing any oil. This plume of smoke and stink only lasts a second and then is gone, but it happens almost every time I start the truck. There is a small coolant leak because I can smell that telltale syrup-like odor as I am driving. Could the coolant leak have anything to do with the startup exhaust smoke and burning smell, or could it be something else? Should I worry too much about it, or is it some sort of annoyance that I will have to live with because it will cost more than it is worth to fix? Any advice would be greatly appreciated. Thanks

@JLStrickland oil burning is blue smoke.
White smoke is coolant.

I suspect you may have leaking intake manifold gaskets. That could allow coolant to enter the combustion chambers. It could also allow coolant to enter the crankcase.
Do you have the 4.3 V6.
There’s an S10 with the 4.3 at my shop right now. The intake manifold gaskets leaked in such a way that coolant went into the crankcase. My colleague replaced the gaskets, but now the engine is knocking. So I’m assuming the bearings took a hit.
You should get your truck checked out quickly.

Look into the coolant reservoir and the radiator. Does the coolant look like oil got in there?
Pull out the engine oil dipstick. Does the oil look like cloudy? Does it look like coffee or milkshake?

I’m not saying you have this problem, but you might. GM V6 and V8 engines in the past have commonly had leaking intake gaskets.

Are there any puddles in the cab?
Is the inside of the windshield cloudy or sticky?

You should have a mechanic do a coolant pressure test.