I have a Free Spirit 10 speed, and while it's not as cushy as my more relaxed bikes, it certainly isn't uncomfortable.

Make sure there's enough air in the tires. If they're low, they'll transfer bumps to the wheels more easily. Fill them up to the max pressure on the sidewall, and then slowly remove air until it feels right.

Go for a 1 1/4" width tire. Cheng Shin (CST) makes a great one in this size that's dirt cheap.