To gather the 7 Dragon balls you can get them randomly through the mail (rare) or grind them through parallel quests. When you have the 7 balls go to the podium in the area where the giant gears are. Shenron will appear and grant you a wish.
The dragon balls are dropped by time patrollers that will appear just before you need to travel through a warp gate. They seem to be able to appear on any quest that is across multiple maps. Before you leave an area press left on the d pad to access your scouter, a time patroller will be easy to spot. Fly over to him or her and press o to start fighting. Once defeated they will either drop 2x material items or 1x material item and 1x key item. The key item is a Dragonball, if you get 2x material item you may want to restart the quest.
I have personally found the time Patroller's easy to find in these quest's.
After you beat Krillan, Tien and Yamcha the patroller will appear in the building to the right.
After you beat the Saibaman and Raditz a time patroller can appear to the left of a warp gate.
As soon as you start you will see the patroller straight in front of you with the scouter.
It's worth noting that 7 wishes are required for the platinum trophy. 2x ultimate attacks, 2x special skills and 3x new characters. - credits to the gamefaqs community for this part