First, I recommend a book called "Zinn and the art of bicycle maintenance".

Second, I'd post your question in the "Classic and Vintage" forum. There's some real Schwinn experts in there.

From a strictly economic standpoint, your best bet is to find a more modern used bike. I did something kinda sorta similar to what you're contemplating and I learned as I went along. It helped that I had access to two LBSes with a huge used parts inventory. Still, even with all the used parts I probably spent $200 and with that $200 plus what I could have sold the bike for, I could have ended up with something better with much less work involved.

I know your bike has some sentimental value and upgrades can also be fun, so it might be worth it to you. Some quick upgrades that might get you going enough to see if you want to continue would be to keep the same number of gears you have but give yourself a wider range. You can do that by replacing your small chainring (one of the gears up front) with one that's got fewer teeth. You can also replace your freewheel (the cluster of gears in the back) with something that has more low end range (the large cog will be larger).

Take your bike to the C & V shop you mentioned, tell him what you'd like to do and have him help you pick out a compatible chainring. Ask him what tools you'll need. He might be willing to remove your old freewheel for free so you might be able to get by without any tools for the moment.

As far as your old bike lasting, I don't think that should be an issue. Bikes can last a long time.

Will this bike be the one to help you hit your stride? Maybe by swapping a few gears you'll be able to figure that out. Schwinn made a few really nice lightweight models but they also made some tanks, -which will last forever but are pretty dang heavy.