There is different care and instructions for different piercings and healing time varies. There is greater risk of infection if you go swimming while the piercing is healing. What I have read, is that you should avoid swimming for two months after getting belly piercing done. It says if you absolutely have to go swimming then disinfect it after. Since you didn't say what type of piercing you got I have posted some links with healing times and more info.

http://www.allpiercings.com/shop/swimming-with-new-belly-ring.html
http://www.navelrings.biz/body-piercing-care.html
http://lifestyle.indianetzone.com/bodyart/1/risks.htm

A Piercing is an open wound, Until the piercing is healed submerging a piercing in a water system that you can not control is risking the piercing becoming contaminated. There is no clear answer to when a piercing is healed cause it depends on how healthy you are and how quickly you in fact heal. Most professional piercers will give you an average healing time and though it is often the longest period possible it is a good rule to go by.

On average here is the break down
above the neck piercings - 8 to 12 weeks
tongue piercings - 2 to 4 weeks
nipple piercings - 24 to 36 weeks
navel piercings - 24 to 52 weekw
genital piercings - 6 to 12 weeks

The worst case is submerging the piercing in natural waters, second would be hot tubs and third would be swimming pools. In most situations if it is a public pool or hotel pool you shouldn't have a problem because they are regulated and tested by local governments. So if you have to go swimming and have a healing piercing, I would suggest a public or hotel pool.

Vaseline is a really dumb idea and could in fact increase your likelihood of problems and prolonged healing. It is a petroleum based product!!! Bad idea. Water proofed bandages are a better optioned if you must going into natural waters.