This is a huge and important topic and it is important to remember that there are many different ways to increase feelings of spirituality in your life. This can be achieved through conventional religion but organised religion need not be central.
It is important to bear the following in mind:
Central to spiritual experience is the shrinking of the self. This can be achieved through a variety of different routes such as a sense of duty and service whereby your own individual needs are subordinated for the good of others or the community. It can also be achieved through meditation, or mindfulness, or different types of practice.
A sense of the sacred is also central to spirituality. This means setting apart from the ordinary and treating with reverence.
The search for meaning, communion and connection is also central to spiritual experience. This can be found in joint pursuit with others or in a strong sense of connection with the natural world and the mysteries of the universe.
The sense of being on a higher moral plane, in touch with our higher selves is also important.
Positive emotions which connect us to others - joy, compassion, forgiveness, gratitude, love - are also at the core of spirituality.