Nothing at all wrong with those shifters or the brake levers. If they work, they work. Just get a bit of steel wool and oil on them so they don't look rusty anymore. Then, get a pair of cone spanners and regrease the hubs. The cone spanners are a good investment because your boy can do his next bike's wheels as well. Tools are a better purchase than spares.
You could get trigger type shifters but the only 6 speed ones are collectors pieces now. Or you could convert to a 7 speed freewheel but they're not good, and a new block will cost as much as the bike did.
Then, new cables, brake blocks and (real speed freak trick coming up!) slick tyres. If you put slicks on that bike it will be considerably faster.
Replacing major components is going to cost more than the bike.