In the words of Dirac: "A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured."
The state of a quantum system is described by some superpostion (linear combination) of states in whatever basis you choose. For discussions about momentum it is convenient to use the basis of states that are eigenstates of the momentum operator. Now, a wave packet will be a superposition of momentum eigenstates, and the uncertainty in the momentum is given by the width of the wavepacket. However (and this is the crucial point), when you make a measurement of the momentum, the system will jump into one of the eigenstates of the system and the measurement of momentum will be the eigenvalue that corresponds to the given eigenstate. So making the measurement on the system actually changes the state of the system! After the measurement is made the system is no longer in the superposition of states (which had uncertainty in the momentum) but instead is in an eigenstate of definite momentum with no uncertainty. Any further momentum measurement will give this same value.
In addition to this you should know that on performing a measurement the probability of obtaining a particular eigenvalue is $|c_n|^2$, where $c_n$ is the coefficient of the particular eigenstate in the linear superposition.
The above forms the very core of quantum mechanics and is known as the superpostion principle and holds for any physical observable. It is covered in detail in all good quantum mechanics texts, I particularly recommend section 1.4 of Sakurai's "Modern Quantum Mechanics" where I found the Dirac quote.
I think the difficulty you are having in your understanding comes from the contrast in the idealised formalism given above where momentum eigenstates are delta functions, and the reality of real world measurements that can never measure exact values of such continuous observables. In the real world a particle will always be represented by some form of wave packet with some spread in position and momentum. It is sensible to define the position and momentum as the central values of these distributions and be aware that there is some uncertainty in these values. The more accurately we measure one, the more the uncertainty in the other will increase. Whenever a more accurate measurement is made, the state of the system changes and the wave packet becomes more and more localised (in position or momentum space). Future measurements will then be more likely to produce values in this localised region.