Azbean,
Are your 3 miles relatively flat? If you were doing fine on a three speed that maybe wasn't even your size or adjusted to fit you, you should consider a coaster brake single speed. Sounds like you like the idea of bicycles and a bike like this-

will serve you while you contemplate a "nice" bike. Zero maintenance. This is at City Grounds in Long Beach CA. But they are everywhere. It reminds me of my Western Flyer after I took the fenders and chain guard off in 1969. Wish I still had it.