DO

Wash your hands with antibacterial soap before cleaning or touching your tattoo.
Keep your bandage on for at least 2 hours. It can be left on overnight.
When removing your bandage, rinse tattoo with warm water and clean paper towel.
Wash twice a day with just warm (not hot) water or a mild fragrance-free liquid soap, and apply a thin layer of ointment (Aquaphor) for the first 2-3 days.  
After the first 2-3 days, switch to a mild unscented lotion (Lubriderm).  All natural?  Try coconut oil.  Avoid lotions with fragrances or aloe.
If possible, use freshly washed bedsheets, clothes, and towel during healing process.  Disposable paper towels are great for direct contact with the tattoo when washing or drying.  

DON'T

Do not scratch or pick. This removes ink and leads to infections!
Avoid direct sunlight. For the first two weeks try to limit sun exposure - if you know you'll be out in the sun for prolonged periods, wear a thin layer of clothing over the tattoo.  After the first two weeks, use a high SPF sunscreen on your tattoo when in the sun. Avoid tanning beds for at least a month.
Avoid soaking in any waters (lake, ocean, bathtub, pool) for the first month, especially critical during the first two weeks.
Avoid heavy sweating in the area of your tattoo for two weeks. Wash after sweating.
Do not let your tattoo stick to sheets or clothing. If stuck, soak the cloth to remove rather than ripping it off.  This is usually only a concern for the first night when it will be a bit oozy.

EXPECT

Color and plasma may seep to the surface, especially when first removing the bandage. This is normal. Wash gently with warm water and wipe with a clean paper towel and reapply ointment or lotion.
For the next week the tattoo and area around it may be slightly swollen, red, and sore.  
Within a few days to a week, a thin scab should form evenly over the tattoo and peel like a sunburn. (Do not pick at it!)  
The color may seem dull or uneven while the tattoo is healing.  

It should be completely healed in one month!  Enjoy your tattoo forever!  For best results, avoid sun exposure (wear SPF), and keep moisturized.