What should I expect with a new tattoo?

A new tattoo is an injury to your skin. It's a lot like a scrape.
Typically, when the artist finishes, he or she will cover the new
tattoo with some sort of bandage. In the old days, this might have
been gauze, but now we have non-stick bandages that are much better.

Some will use clear plastic sheet, such as Saran Wrap (TM). This is
all right, and has the advantage that you can look at your new tattoo without having to lift the bandage. The bandage is largely to protect the tattoo and your clothes, since new tattoos ooze a lot.

Also note that tattoos will ooze ink along with the other fluids, and it is not unusual to find an imprint of your new tattoo on the inside of the bandage with you take it off. This is normal. Also, it is normal for the oozing to go on for about 24 hours.