As you already suggested, it takes time. I would say that to begin to be a good piano tuner takes at least 3 years and you still have plenty of room to improve.
It helps to have a good and discerning ear, but you do not need what people imprecisely call perfect pitch. You will need a good reference tuning fork or pro electronic tuning reference. I prefer using a C tuning fork but most people use a A.
One of the most difficult part for amateurs is that they usually have only one piano to tune, their own and that
they usually have been very influenced by this piano and its probable incorrect tuning evolving in time.
they do not know what they want in term of temperament
they are not aware of the numerous pitfalls of our human sense of sound.
they believe that all pianos are more or less the same as regard tuning
A thing you must do and that you must do often is to go in as many places as you can (piano sellers, friends, etc.) and hear different pianos to help develop a feeling of the very diverse compromise that a tuning is.
You need some basic tools :
a tuning wrench matching the model of your piano. Most upright pianos have the same kind. Choose one with a solid and long handle, it is more comfortable.
tools to mute or isolate strings (when tuning notes having 2 or 3 strings, also initially to have a whole central octave with only 1 active string by semitone)
and a lot of patience. The best would be to have two different pianos in the same room. One more correctly in tune than the other. Practice on one trying to match the tuning quality of the other.
EDIT: Further basic advices as several suggested I extend my answer.
There is a lot of easy things you can do as a beginner. Do not rush.
On a piano which is not too out of tune, concentrate on a few notes in the medium part of the piano (extreme low and the upper register can be tricky at first) that are the most out of tune or the most disagreeable sound. This is usually because the two or three strings of this note have been drifting from each other. First mute the (two) lateral strings on the target note and do the same to a note one octave apart. First tune the central string using the octaved note as a reference. Then unmute one of the lateral ones and match it with the central string, listening for beats, try to always finish your tuning by a smooth move pulling the string while playing (it is usually more stable). re-mute then match the other lateral one with the central one. Unmute all and check first as an isolated note and with the octave.
Most amateur pianos are not tuned sufficiently frequently, especially in their early years, that's a pity because it degrades their further tunability. If yours has drifted a lot from the reference tuning you want (usually A=442Hz or A=440Hz), do not try to re-tune it at once or in one move. First learn to improve its harmonization : improve individual notes, try to determine how the drift is spread other the piano (it is quite common that the medium has drifted more than the bass or the treble parts because of a more frequent usage). Look closely at the strings and tuning systems: are all strings and systems in good shape (no rust, regular winding, no excessive pinching, even outside length of tuning axes, etc.) ?
Good tuning is associated with good key subsystems : learn more about the mechanics, look at the hammers (are they moving silently, are they aligned with the strings angle, are they moving back in place quickly ?), at the felt (is it even from a key to another ? is it too marked by the strings ?).
As promised. One good (but not as complete as it claims) reference in English, now in paperback is :
Piano Servicing, Tuning, and Rebuilding: For the Professional, the Student, and the Hobbyist.
Arthur Reblitz, 1997, 1-8795-1103-7
Beware, the same author made several books on piano construction, mechanical piano, history of piano-making.
There are tuning exercises and beat charts for a common tuning system.