There are different levels of Buddhists, and everybody takes and uses as much of the Dharma as they want. The first thing is to take refuge in the Three Jewels, then on you are a Buddhist, no matter what. If you can keep, or aspire to keep, some precepts and vows, that's already part of the practice. And when I say practice, it means practising the Dharma in one's life, and not in a narrow sense of sitting on a cushion.
From a fundamental perspective, here is a collection of teachings addressed to lay people: Lay Buddhist practice
. This discourse is highly recommended: Sigalovada Sutta - The Layperson's Code of Discipline
And if we are talking about Mahayana, the six paramitas are easily applicable to lay life. It includes methods to relate to others (giving, discipline), central qualities to nurture (patience, effort), and inner methods (meditation, wisdom). If that sounds too much, it can be reduced to a single practice: compassion. Devotion is another important part of Buddhism, and you can see its tangible presence in any Buddhist country. One part of that devotion is "mindfulness of Buddha", that is practised in the form of recitation and prostration. The central method of the Pure Land school is being mindful of Amitabha Buddha, and it is the most popular among both the laity and the monastics. But if you feel more connected to another buddha or bodhisattva, that's fine too.
If you are more drawn to Vajrayana, it has a lot to offer too. As an example: The Pointing-Out Instruction to the Old Lady
I think that even for a busy lay person like you there are many many options. You just have to choose.