What are the Hebrew or Greek words for joy and rejoice?

The two Greek words used -- chairein and chaironton -- are both forms of the same Greek word: chairo. It means 'to rejoice, be glad, full of joy.' The first is a Present Active Infinitive, the second is a Present Active Participle (genitive case, masculine, plural). This word is related to charis which means 'grace.' In its broader meaning it is a form of greeting, such as 'be well' or 'thrive.'

http://biblehub.com/greek/5479.htm
http://biblehub.com/greek/5463.htm

Hebrew has several words for joy, each with different shades of meaning.

'Simcha' (ch is a soft gutteral, between h and the ch of 'Bach') means happiness in the most full sense.
'Gila' is a stronger sensation of joy but more transient and worldly (related to 'gal' = wave.)
'rina' is a joy related to singing and shouting.
'ditza' is joy related to dancing.
'hanaah' enjoyment (of something specific.)

http://www.theopedia.com/joy
http://biblehub.com/hebrew/1523.htm
http://biblehub.com/hebrew/8057.htm