To do calculations using Tsiolkovsky's Rocket Equation, you need to know the velocity of the exhaust gas of the rocket. Here's a basic way of doing that for small rockets, at least.
Weigh the rocket before the experiment. Attach the rocket to a force measuring device.
Fire the rocket and start the stopwatch. Observe the average force F it generates.
When the rocket stops burning, note how much time it took. Weigh the rocket again.
Force = rate of change of momentum $$ F= \frac{dp}{dt} = V_e \frac{dm}{dt} \approx V_e \frac{m_2 - m_1}{\Delta t}$$
Where m_1 and m_2 are the intial and final mass of the rocket, F is the measured force, and \(\Delta t\) is the amount of time the rocket is fired.
Rearranging it a bit: $$ V_e \approx F \frac{\Delta t}{m_2 - m_1} $$
This gives us an indirect and low tech way of estimating the exhaust velocity and thus specific impulse of the rocket.
Oh, and measuring the thrust force F is useful too.
This is assuming thrust is constant, of course. It won't work for things like compressed air, or water rockets. Something that can do constant sustained burn for like ten seconds, maybe.
Negative number for exhaust Velocity? Of course it is, it's directed backwards!
Japans's space program JAXA started with small scale experiments similar to this. This link describes the "Pencil Rocket", so named because of its size. Actually it's much bigger than a pencil, but still really tiny for starting an aerospace program.
Sometimes starting small, building a strong team and institutional knowledge over years can yield impressive results. This eventually led to the modern Japanese space program.
This video shows a rocket test stand for a small motor.
Did you know there was a Lebanese Space Program?