You can get 700 X 28c tires on there ok. It will make a bit of difference when you are going up hills. If you keep your tires properly inflated they will be about the same on flat roads.

There are 'touring' flat resistant tires and there are lighter 'performance' tires that make another small improvement, but still not too much. Lighter rims would enable you to put on narrower tires still and possibly use lightweight racing tires. The performance difference would be a little more noticeable, but still not night and day.

Also, smaller tires have a smaller circumference you may have to use a bigger gear when you ride, or spin your legs faster. And lightweight tires generally have more flats.