illustration (attribution, if any possible, is at the end of the article)
The career of a bodhisattva starts, strictly speaking, when one commits to bodhicitta. It comes after some buddhist understanding though: once one has understood (at least conventionally) dukkha, and when one has (partly) loosened the grip of selfishness, one might wish for others to be free from sufferings.
One could say that bodhicitta is born from compassion (wishing all beings to be free from suffering).
Once one has enough compassion but also has started to cultivate faith / trust / confidence in the Dharma, one reaches the conclusion that the best way to help all beings is to strive to become a buddha, to strive to Awaken.
No matter how dedicated one is, a focus on relieving phenomenal pain is meritorious but will never reach beyond the 'local' population; only transcending pain altogether as well as transcending ordinary limitations might allow one to help all beings.
Buddhahood is the only option to really help all beings, because fundamentally it is the only form of engagement that addresses the causes of suffering, rather than merely the symptoms.
Bodhicitta may, in some cases, arise 'spontaneously' (usually due to karma in fact) but one can also cultivate it systematically.
It is born from compassion. Compassion can aim at three different levels (beings, factors & aggregates, objectless (or 'empty' beings)).
The first level of compassion is enough to support bodhicitta.
Compassion itself can be developed via equanimity and loving-kindness (having no enemy and wishing others to be happy). These two can themselves be developed, and Buddhism provides many guided meditations to achieve so.
Of course, as it happens, the reorientation from "buddhahood for one's own benefit" (selfish search for cessation of individual suffering) towards "buddhahood for all beings" is a transformative gate in the bodhisattva's career. It becomes an uplifting aspiration, which may prove enough to step out of apathy or depression. Once the resolve is taken, one becomes a true "beginner bodhisattva", a child of the buddhas.
Similarly to Angulimala (an ex serial-killer!) being instructed by the Buddha to state to the woman in labour "Sister, since I was born I do not recall intentionally killing a living being. Through this truth may there be wellbeing for you, wellbeing for your fetus", the aspiring bodhicitta transforms even the worst being into a child of the buddhas and it plays an important part in maintaining the motivation to cultivate the necessary qualities.
The ten bhūmis (stages of the bodhisattva career) might seem extremely difficult to attain and any supportive inspiration might play a role in one's actually reaching the goal.
Initially, of course, bodhicitta is likely to be very 'conventional' (e.g. still imagining "beings to save" as independent from oneself, imagining saṃsāra to be separate from nirvāṇa, etc.). Only via the practical or applied version of bodhicitta, the cultivation of the six perfections, will one transform bodhicitta from being a "purpose" into the 'ultimate' buddha-mind beyond duality, into thatness.
photo: "Seated Guanyin (Kuan-yin) Bodhisattva" © The Walters Art Museum (http://art.thewalters.org/detail/4483)