My engine smokes at start up

If that is blue smoke, and it clears up after a minute or two, the most common suspect is deteriorated valve guide seals. They will allow engine oil to seep into the cylinders. This will happen very slowly when the engine is warmed up and running, so you will not notice it. When the engine is stopped and the oil is still very fluid, it will run down the valve stems past the seals, and collect in the cylinders. It burns and goes out the exhaust when the engine is restarted.

The fix for this is to have the valve stem seals replaced. Often this can be done without removing the cylinder head, but on most Toyota engines, special tools are needed due to the design of the head, and those tools only work with the head off the engine.

This also becomes a temporary problem when the oil becomes diluted with raw gas during a crank/no-start problem caused by loss of spark. The oil must be changed as soon as that no-start problem is solved, otherwise the gas will prevent the oil from separating moving parts from each other and the friction will cause very serious engine damage.

This can also become a bigger problem when the additives in the oil wear out. Typically this happens when you go too long between oil changes. When the oil is changed, the additives are replenished. Those include viscosity index improvers, detergents, and seal conditioners.