When taking my own blood pressure, I hear a few weaker beats and then
A strong one. Do I record my blood pressure as the first weak beat I hear, or
The first stronger one? Thank you.

Since you are measuring when you first hear sound and when you no longer hear it, it doesn't really matter. They should be very close together, a second apart. It's really an approximation and won't be exact anyway. There are many variables - the way the cuff is wrapped, how well you hear, how good the stethoscope is, what time of day it is, what you have been doing...

Just take note of the reading, that is all you have to do

I gave up trying to measure with a cuff and stethoscope. You can buy decent digital sphygmomanometers for $30-40