We have a split level that was built in 1970.
the family room has two big square recessed lights that hold one 60 watt light bulb each.
This is inadequate. The ceiling has older radiant heat. I want to replace the lighting but I am not sure what to use. Does anyone have an ideas on what to use because the squares are about 10 inches square. Any help would be appreciated.

Quickest thing to try is replacing that 60W incandescent bulb with a brighter compact florescent bulb.

You probably can't go bigger than 60W incandescent in that recessed fixture, because of heat problems. But CFL bulbs run much cooler. You can probably put a CFL bulb with the equivalent light of a 150W or 200W incandescent in that fixture with no problem.

So try that first. If it provides enough light, you don't have to do any rewiring or replacing fixtures or anything like that.