Re: Good nature trails in Columbia?
I second Congaree National Park. It's about a 30 minute drive from Columbia, but it's a cool park with some interesting trails.
Peachtree Rock is a decent little place to "hike" too, but it's probably 45 minutes atleast from Columbia.
I say "hike" because I have difficulty considering walking in the SC midlands "hiking".
disclaimer: I know nothing about the dog rules at these places.