I would recommend that you go to see either your primary care doctor
or a dermatologist
about this spots, as they will be able to confirm the diagnosis for you and help you decide on a treatment.
Given the way you describe how these white spots developed, namely that they became prominent after you got a tan, I think the most likely possibility is something called tinea versicolor. Tinea versicolor is a fungal infection of the skin. It causes light colored patches which often appear on the shoulders, back and chest. These patches are made more obvious by tanning, as they do not tan; this has led to one common name for tinea versicolor, "sun fungus."
Fortunately, if this diagnosis is confirmed, the condition is relatively easy to treat with prescription antifungal creams or lotions, which your doctor will be happy to provide for you.
Your doctor will also be able to consider alternative causes of these white spots, such as damage to the skin from sun burn, dry patches of eczema
, and the like. While you are waiting to get in to see your doctor, make sure that you keep your skin well-protected and use plenty of sun screen!