The rate at which the balloon rises is useful to know. It allows you to predict when and where the balloon will come to float. If you know when and where you will launch, there are websites that will tell you where you will come to float and which way you will be going.
If you have an SBS balloon, they give the rate in their instructions. If you have a chinese balloon, you need to assume something. You could just use the SBS numbers and be close. If you want to get better data, it can be done.
You can use an app like VidAnalysis or some other to video the release of a balloon. It can be done indoors with a high ceiling and the appropriate weight attached to the balloon. With a video, you can pinpoint the same spot and graph the ascent. The balloon will stabilize within a meter or so and there you have a measurement. This can be done with several weights attached to the balloon to verify how ascent is affected by free lift. A great classroom project.
The rate of ascent of the balloon is essentially constant until the balloon reaches full inflation and then it will slow down until it reaches float. Since the balloon is travelling slowly, the velocity is directly proportional to the free lift. The flow around the balloon is laminar and that is different from airplanes where the drag and velocity are related to the velocity squared. Recall that the lift is constant until the balloon is fully inflated.
Once the balloon is inflated, the velocity will decrease linearly with altitude until it reaches float. This resembles the discharge of a capacitor and the time constant is the distance between full inflation and float divided by the rate of ascent. It takes about three time constants to come to float. This will not matter much since it will be in the order of 5 to 10 minutes but it is comforting to know it is done.