One day is plenty of time to see all the attractions in Cullman. The best days to visit are Monday through Friday, as the Cullman County Museum is closed on Saturday, and the Shrine and the Grotto are mostly religious sites which are inappropriate on Sunday.
You should begin your tour just outside of Oneonta in Blount County. Here you can see the covered bridges. Depending on your interest, and how many times you drive back and forth on them, you can see all the bridges in about an hour (free). If you stay the night in Cullman of Gadsden, you can leave either at 6:30am to be at the bridges by 7am.
Next, head to Hanceville, to the Shrine of the Most Blessed Sacrament (free). If you see the bridges by 7:30am, you should make it to the shrine by 8am. Visit the Castle first to get the guide to the site. Follow the guide around the site to see all of the interesting features. It takes about 2 hours to see, and appreciate this site.
Drive to Cullman and visit the Ave Maria Grotto ($5). Spend a couple of hours exploring the grotto.
Move to the Cullman County Museum ($2). Touring the museum should take about an hour.
Next, drive to downtown Cullman, here you will be able to spend a couple of minutes looking through the many antique shops, or grab a bite to eat at one of the local restaurants. You might also check out the Historic Warehouse District, which houses a multitude of unique shops. Also, you might want to check out A Touch of German which offers a selection of German wares.
While walking around downtown Cullman, take note of the Weiss cottage (the oldest building in town), the Cullman depot, and the historic district. You should plan on leaving Cullman by 2pm.
Leave Cullman heading west and you will run into the Clarkson Covered Bridge. There is little here except the bridge which serves as memorial for the battle that took place here. Touring this site will take less then half an hour.
Next, head to Double Springs. Here you will find the "Free State of Winston" statue. This memorial is here to celebrate the fact that Winston county actually succeeded from the state of Alabama, right after Alabama succeeded from the Union before the state of the Civil War. Forced to comply by surrounding Confederates, this area was known as a haven for those who were sympathetic to northern issues.
Continue traveling west to the Natural Bridge ($3). Touring this site will take less then half an hour.
Finally, continue traveling west to Hamilton for some shopping, or a nice dinner. You should see Brown's Pottery for some interesting handmade pottery, sculpted in the historic fashion. Be sure to see the face jugs, which are the shops specialty.
You should be complete with all of these attractions by 5pm.
Total cost of the trip is $10, plus food, gas, and lodging.
I recommend a day trip to Cullman for anyone interested in spirituality and for families with children. Also, this trip is nice for frugal travellers as all the attractions combined cost only $10; hotels are cheaper as this is not a major tourist destination.