Some fingering charts are easy to read, and some are not. Find one that is as easy to understand as possible and give it to your students.
Fingering for brass instruments is simple, so students can learn it in a few days. If instructors thought it was natural to memorize it in a few days, they wouldn't have students write numbers on the music.
The contents of the fingering chart are the same as the Harmonics Chart, but rearranged in order of pitch. I have written a Fingering Chart that makes it easier to understand the relationship with harmonics.
Among brass instruments, double horn students should learn the fingering for both the F and B♭ horns. double horn.
Fingering for woodwind instruments will take a little time.
I cannot write fingering charts for Wood Wind instruments other than the bassoon here (you can download a PDF file of the bassoon fingering chart below), but many instruction books for the flute, saxophone, and clarinet come with easily illustrated fingering charts. There are also several fingering charts on the Internet.
The flute has a range of three octaves, and for about two octaves, plays should control the angle of the air to play notes above and below the octave.
The saxophone usually has a range of about two and a half octaves, and for the lower two octaves (Except for the lowest range), you can play notes an octave above and below with the same fingering, except for the octave keys.
The clarinet has a range of more than three octaves, and only the left range is used in wind ensembles, but due to differences in the structure of the instrument, fingerings do not match for each octave. Among them, fingerings for each note (notes in the same measure in the table below) from the lowest note E to the middle note F
and from the middle note B to the high note C
are the same except for the left thumb key (on the clarinet, this key is called the register key). 
For the clarinet, the upper and lower notes in same measure above are the same fingering except for the register keys.
For basic fingering rules that students who are just starting to play the clarinet should first learn, please see this page.
Fingering for the bassoon is much more complicated than other woodwind instruments, as the number of keys varies depending on the instrument and the basic fingering often results in poor intonation (the optimal fingering varies depending on the instrument).
It may be difficult to remember the fingers just by using the fingering chart because there are so many keys, but I have created a fingering chart (prototype PDF file) that diagrams the keys of standard instruments as much as possible so that it is easy to understand for first-time teachers and students.
The bassoon has a wider range, but here I have written the lowest note Bb to the highest Bb. I don't think elementary school students need to even go up to Bb.
Blank bassoon fingering chart (PDF file)
Give each player a fingering chart for each instrument so that they can refer to it whenever they need to.
Students should be taught to watch the chart over and over again until they have it memorized, and the teacher should not teach them anything except for special fingerings.