My suzuki step 125 has a lot white smoke smell oil

White smoke usually means burning coolant in combustion chamber and lots of it is not good. Check for oil in radiator coolant reservoir. If oil is present your engine is bad.

Leaky injector

Are you losing coolant?

Blue smoke is oil,white smoke steam or trans mission fluid. Check that auto fluid not going up a vacuum line if it has a modulator.

white were? the exhaust tip, or other spot, long pipe, name spot?
white, is steam or OIL.
oil smells like burnt oil and takes for ever to dispate
steam is water, and smells , sweet due to 50% glycol inside.
and goes away fast.
but all Gas engines make water at all the time running as a by product of combustion and for sure on cold days, some times full time cold
due to climate effects.
does your neighbors car smoke at same time, (look)
bingo , dew point.

all gas engine smoke cold started, (steam)

so first find out what it is,
take the dip stick out of a hot engine. stalled, drip it on the exhaust
see that smoke ,smell it? that is burning oil
now you know what you have, steam or oil. smoke.

White SWEET smelling smoke, or white OILY smelling smoke? If it's white thick SWEET smelling smoke, you have a blown head gasket - that's the smell of antifreeze burning in the engine, which it should NEVER do. If it's white OILY smelling smoke (smells a lot like diesel), then you're getting too much fuel into the cylinders.

If it's sweet-smelling smoke (antifreeze), pull the oil dipstick & check the oil level. If the oil level is HIGH and the oil looks like a mocha milkshake, that's absolutely conclusive - that's what happens when antifreeze gets into your oil, and it's nasty corrosive to engine parts.

I'm not at all surprised that the oil light is on if it won't start - you won't get much oil pressure until the engine is running.

White or Blue smoke? Have you checked the oil to see if there is any water in it?

White Smoke -- Coolant - bad head gasket

The key is the color and smell of the smoke.

IF coolant, the smoke is REALLY white and usually lots, and this smoke has a sweet smell. A pressure test may show some loss, and as well, often you get a little misfire that goes away.

If Oil, the color is white, but with a lot of blue, or darker...

Most of the time, after the converter warms up, it eats and converts both types of issues...

Steam, can be normal. Remember that for every gallon of fuel you burn, you actually generate 1 gallon of water. Typically evaporates or exits as steam, and is barely perceptable.

if it's a 2001 model it is fuel injected which means it doesn't have a carby. the white grey smoke is burning oil which means you might want to keep an eye on you're oil level but the strong smell of fuel and lack of acceleration could be caused by a fuel leak somewhere