Depend on the method of breath there are various practitioners. for ex. 'Pranayama' in hindu style for health, Buddhist use 'Samatha' is for calm and 'Vipassana' for insight.
In 'Samatha' there are five(or 4) stages of ruupa jhaana and four of aruupa jhaana.
For 'Vipassana' the stages are shown in
Aṅguttara Nikāya - The Book of the Tens - 60 Girimānanda
On one occasion the Blessed One was dwelling at Sāvatthī in Jeta’s
Grove, Anāthapiṇḍika’s Park. Now on that occasion the Venerable
Girimānanda was sick, afflicted, and gravely ill. Then the Venerable
Ānanda approached the Blessed One, paid homage to him, sat down to one
side, and said to him:
“Bhante, the Venerable Girimānanda is sick, afflicted, and gravely
ill. It would be good if the Blessed One would visit him out of
compassion.”
“If, Ānanda, you visit the bhikkhu Girimānanda and speak to him about
ten perceptions, it is possible that on hearing about them his
affliction will immediately subside. What are the ten?
“(1) The perception of impermanence, (2) the perception of non-self,
(3) the perception of unattractiveness, (4) the perception of danger,
(5) the perception of abandoning, (6) the perception of dispassion,
(7) the perception of cessation, (8) the perception of non-delight in
the entire world, (9) the perception of impermanence in all
conditioned phenomena, and (10) mindfulness of breathing.
(Pali: Katamā dasa? Aniccasaññā, anattasaññā, asubhasaññā, ādīnavasaññā,
pahānasaññā, virāgasaññā, nirodhasaññā, sabbaloke anabhiratasaññā,
sabbasaṅkhāresu anicchāsaññā, ānāpānassati.....)
According the sutta before starting ānāpānassati(mindfulness of breathing) meditation, one should develop other 9 perceptions.
For ex. nirodhasaññā (the perception of cessation)
( ‘etaṃ santaṃ etaṃ paṇītaṃ yadidaṃ sabbasaṅkhārasamatho sabbūpadhippaṭinissaggo taṇhākkhayo nirodho nibbānan’ti)
‘This is peaceful, this is sublime, that is,
the stilling of all activities, the relinquishment of all
acquisitions, the destruction of craving, cessation, nibbāna.’
So these are the 'perceptions' that to focus on between two breaths.
(But practically to 'vipassana samapaththi' those perceptions already gains. so let quit the breath. Do not try to control, let it gone. If the vipassana perceptions are not developed enough strength, one can fell to sleep. Better to have find a experience teacher.)
The detail stages in developing as follows
(10) “And what, Ānanda, is mindfulness of breathing? Here, a bhikkhu,
having gone to the forest, to the foot of a tree, or to an empty hut,
sits down. Having folded his legs crosswise, straightened his body,
and established mindfulness in front of him, just mindful he breathes
in, mindful he breathes out.
“Breathing in long, he knows: ‘I breathe in long’; or breathing out
long, he knows: ‘I breathe out long.’ Breathing in short, he knows: ‘I
breathe in short’; or breathing out short, he knows: ‘I breathe out
short.’ He trains thus: ‘Experiencing the whole body, I will breathe
in’; he trains thus: ‘Experiencing the whole body, I will breathe
out.’ He trains thus: ‘Tranquilizing the bodily activity, I will
breathe in’; he trains thus: ‘Tranquilizing the bodily activity, I
will breathe out.’
“He trains thus: ‘Experiencing rapture, I will breathe in’; he trains
thus: ‘Experiencing rapture, I will breathe out.’ He trains thus:
‘Experiencing happiness, I will breathe in’; he trains thus:
‘Experiencing happiness, I will breathe out.’ He trains thus:
‘Experiencing the mental activity, I will breathe in’; he trains thus:
‘Experiencing the mental activity, I will breathe out.’ He trains
thus: ‘Tranquilizing the mental activity, I will breathe in’; he
trains thus: ‘Tranquilizing the mental activity, I will breathe out.’
“He trains thus: ‘Experiencing the mind, I will breathe in’; he trains
thus: ‘Experiencing the mind, I will breathe out.’ He trains thus:
‘Gladdening the mind, I will breathe in’; he trains thus: ‘Gladdening
the mind, I will breathe out.’ He trains thus: ‘Concentrating the
mind, I will breathe in’; he trains thus: ‘Concentrating the mind, I
will breathe out.’ He trains thus: ‘Liberating the mind, I will
breathe in’; he trains thus: ‘Liberating the mind, I will breathe
out.’
“He trains thus: ‘Contemplating impermanence, I will breathe in’; he
trains thus: ‘Contemplating impermanence, I will breathe out.’ He
trains thus: ‘Contemplating fading away, I will breathe in’; he trains
thus: ‘Contemplating fading away, I will breathe out.’ He trains thus:
‘Contemplating cessation, I will breathe in’; he trains thus:
‘Contemplating cessation, I will breathe out.’ He trains thus:
‘Contemplating relinquishment, I will breathe in’; he trains thus:
‘Contemplating relinquishment, I will breathe out.’
“This is called mindfulness of breathing.
“If, Ānanda, you visit the bhikkhu Girimānanda and speak to him about
these ten perceptions, it is possible that on hearing about them he
will immediately recover from his affliction.”