Originally Posted by
Mr IGH
Yes, that's the right thought on the headset, I've used many BMX headsets on Schwinn lightweights. You can use the stock Varsity headset and replace the top nut with a 1" headset top nut that's 22.2 instead of .833 (or ream/dremel the stock nut). The fork you picked is a common replacement over on the Schwinn forum, it'll work fine. Niagra Cycles on Amazon might have a better price. Best weight loss will be the rims, they weigh a ton. 700C wheels will fit, you might have to file the brake calipers a little bit to make it work.
I think it's a nice idea to use your Dad's old bike. Great reuse and those old Schwinns ride nice. A few extra pounds is nothing when you're young, hell, your Dad was able to ride it

Thanks for the help! Yea, weight isn't really an issue as I already have a nice roadie that I take around. I just wanted to have a nice project to work on to ride for the summertime and I figured what better than my dad's old summer bike?!