The beach! Steer away from Barceloneta though... it´s much nicer if you walk 15-20 minutes to the beaches further east. There are beach bars dotted around, and the kids love it. You can even look into windsurfing for the 15-year old.
Hiring bikes is a great way to see the city, you can get bike seats, though they may be small for the five year old. There is a great network of bicycle lanes. You can hire bikes by the beach in Barceloneta.
Camp nou... 15 year old boys love it!
The IMAX cinema in Port vell, is good... 3-D films for kids.
The Zoo.. always a safe bet
Walk around Barri Gotic, it´s so beautiful and fun even for kids
Have fun!
Parque Guell is lovely
Sagrada Familia is very impressive, and you can climb up the spires
The museo de sciencias is great for kids
Hire pedalos in Parque Ciutadella
Cable car to Montjuic
Tibidabo fun fair
Las Ramblas, to se the living statues.. they´re touristy but fun (hold on to your handbags though)
Hope that helps... have fun