A piercing is technically a wound so bleeding is normal when the earring is removed, bumped, or turned. Be gentle with a new piercing. Give it time to heal itself over the next week. The entire healing process of the piercing can take up to a year which is why there may be drainage noted with any manipulation/change of earrings.
Definitely keep up with cleaning to prevent bacteria from irritating the piercing. You can clean away the dried drainage with a Q-Tip. (Check to make sure no fibers are being left behind by the q-tip!) You may also apply a warm compress daily to alleviate any discomfort.