How does one become a lay Buddhist? I'm well aware that there are many different types of Buddhism, so I'm looking for the fortune-cookie sized description of the various kinds of induction/process/rituals/ceremony involved for the lay follower.
For Theravada and some Mahayana traditions becoming a lay Buddhist is as simple as taking refuge in the Triple Gem of the Buddha,  Dhamma, Sangha and taking the 5 precepts. This is usually at a ceremony and "given" by the monks however the true taking  of refuge is something done in the mind,  not recited or given.  

You are in short taking refuge, showing trust in the Buddha and his awakening,  the dhamma as the truth,  and the sangha as the keepers of that path,  more specifically the aryan(awakened)  Sangha and also all monastics.  Then you attempt to live a more skillful life that is beneficial to yourself and others by following the precepts to the best of your ability. 

The ceremony usually involves a recitation in the Pali for Theravadan or the cultural language for the Mahayana. 

This link to Bhikkhu Bodhi's article on the subject is great for further reading on the subject. 

http://www.accesstoinsight.org/lib/authors/bodhi/wheel282.html

Going for refuge,  recited x3

Buddham saranam gacchami
I go for refuge to the Buddha;

Dhammam saranam gacchami
I go for refuge to the Dhamma;

Sangham saranam gacchami
I go for refuge to the Sangha.

And precepts:

(1) the training rule of abstaining from taking life;

(2) the training rule of abstaining from taking what is not given;

(3) the training rule of abstaining from sexual misconduct;

(4) the training rule of abstaining from false speech; and

(5) the training rule of abstaining from fermented and distilled intoxicants which are the basis for heedlessness.