I have a Schwinn that I am looking to clean up, I know no recommendations for a Schwinn but a free bike is a free bike.
I've mentioned this in other threads, but fix that damn bike up. It's not the best design. Heavy, inefficient posture, but it's decently put together. Will be less problem-prone than a Huffy over time for sure. (Source: my sister has one and I've checked it out briefly a few times while at her house.)
My backup bike is a Schwinn Mesa MTB that's got at least 2,000 miles on it. Again, not great, but not terrible either.
For something that could be acquired cheaply (used), hybrids like the FX are great. The FX is basically the Honda Accord of bikedom. I have a 10 year old 7500 FX I got for $250. I have pulled 300 pounds with it (for a total load of 500). Fully built up with fenders, lights, rack, panniers I've got close to $600 in it but I've logged 3200 miles and counting. Obviously I've burned through some chains, tubes, and finally had to replace the terribly old (possibly original) tires it came with.
That said, the newer designs have even better geometry, IMO. Despite being way too small for me, I still love tooling around on my wife's Giant Escape 3 (only one year old, and gotten for $280).
I'm patiently waiting for longtail cargo bikes to lose value, but I've yet to see a listing for under $750. Ditto true touring or cyclocross bikes.