Where rockets are concerned, Newton's laws are not the first bit of science to turn to. Rather, as the basis of the Rocket Equation, conservation of linear momentum is better.
Starting with momentum = mass x velocity, for a rocket at rest, the momentum is obviously zero. Start the rocket and propellant/oxidiser/combustion products shoot out of the back. If we take the forward direction as the positive one for vectors like velocity then these substances have a negative momentum. To preserve the zero momentum of our rocket/fuel "system", the rocket must have a positive momentum of equal magnitude, achieved by its mass moving forward with a certain positive velocity.
Over the course of a journey, we have to use calculus since the mass of the rocket is gradually changing due to loss of fuel/propellant.
A rocket, or indeed a turbine engine on a static test can, indeed, be said to have a certain amount of thrust. Stand behind the engine and you will, of course, only be too aware of this. However, in explaining the movement/velocity of the rocket, we do not need to invoke the term "thrust".
SFS