If it just does it a little its probably torque steer. Its common in fwd vehicles.
The tires may need alignment if it pulls to one side and or the wheels may need to be balanced
A car accelerates from to at a rate of How far does it travel while accelerating
Drive shaft is going out
They are
Slow engine RPM is the main reason a car may be sluggish while accelerating. However, it also could be due to the oil.
dirty fuel line
Have your brake system inspected. Sounds like you have a caliber sticking.
it is the person who pulls out on your street
A gas leak with the car sputtering while accelerating could be a hole in the vacuum hose that controls the flow of gasoline out of the tank. This type of leak is usually not caused by the fuel pump itself.
bad wheel bearings
It's possible you have a bad wheel bearing, or, if it's a front wheel drive car, bad CV joints are also a possibility.
In Britain it is the car that pulls out that is at fault. Indicators are regarded only as "courtesy" items and you trust them at your own risk.