What's the best way to do this experiment with little error margins?

Sigh. Do your own homework.

I would read the text and follow the instructions printed there. It was done for your edification.

It sounds like you’re measuring two things:
(1) The mass attached to the spring.
(2) The period of oscillation.

For minimum error:
(1) Use a high quality scales, correctly zeroed before each weighing.

(2) If you are measuring the period by counting oscillations per unit time: choose a longer time period and try to estimate fractional oscillations when time is up.

If you are measuring the period by timing a certain number of complete oscillations: count a larger number of oscillations. Realize that hand-triggering a stopwatch is only accurate to maybe 100 msec or more, requiring more sophisticated means for greater accuracy.

Don’t forget the influence of the earth’s rotation and the gravitational pull of the moon and the sun. You should also correct for declination and create a perfect vacuum. ;-)

The statistician in me would also say, use as many varying mass examples as you can. The larger the sample base, the more accurately you can measure trends.