Which aspects or factors do I take into consideration while I record observations on any celestial object?
There are several observational astronomy tasks that have traditionally had active amateur participation. The important things to note will vary depending on the task, but will include the precise time, the precise geographic location you are observing from, and the celestial coordinates (Right Ascension and Declination) of what you are looking at.
The American Association of Variable Star Observers collects amateur observations of variable star brightness.
Amateur observations of minor planet occultations have helped provide detailed information about the shapes and precise orbits of minor planets, including helping to guide the New Horizons probe.
Amateurs have provided valuable information about transient phenomenon in the solar system, such as impacts on the moon or Jupiter.
The International Meteor Organization supplies a guide for observing meteor showers.
And of course, amateur observers have discovered many comets, although they are increasingly scooped by automated whole-sky searches.