Once I have used freopen, how can I get the original stdout (or stdin) back?

There isn't a good way. If you need to switch back, the best solution is not to have used freopen in the first place. Try using your own explicit output (or input) stream variable, which you can reassign at will, while leaving the original stdout (or stdin) undisturbed. For example, declare a global
FILE *ofp;
and replace all calls to printf( ... ) with fprintf(ofp, ... ). (Obviously, you'll have to check for calls to putchar and puts, too.) Then you can set ofp to stdout or to anything else.
You might wonder if you could skip freopen entirely, and do something like
FILE *savestdout = stdout; stdout = fopen(file, "w"); /* WRONG */

leaving yourself able to restore stdout later by doing
stdout = savestdout; /* WRONG */
but code like this is not likely to work, because stdout (and stdin and stderr) are typically constants which cannot be reassigned (which is why freopen exists in the first place). It may be possible, in a nonportable way, to save away information about a stream before calling freopen to open some file in its place, such that the original stream can later be restored. The most straightforward and reliable way is to manipulate the underlying file descriptors using a system-specific call such as dup or dup2, if available Another is to copy or inspect the contents of the FILE structure, but this is exceedingly nonportable and unreliable.
Under some systems, you might be able to reopen a special device file (such as /dev/fd/1 under modern versions of Unix) which is still attached to (for example) the original standard output. You can, under some systems, explicitly re-open the controlling terminal, but this isn't necessarily what you want, since the original input or output (i.e. what stdin or stdout had been before you called freopen) could have been redirected from the command line.
All of this pertains to stdio redirection initially performed by freopen, affecting the I/O calls within the same program that called freopen. If what you're trying to do is capture the result of a subprogram execution, freopen probably won't work anyway;

How can I redirect stdin or stdout to a file from within a program?

Use freopen. If you're calling a function f() which writes to stdout, and you want to send its output to a file, and you don't have the option of rewriting f, you can use a sequence like:
freopen(file, "w", stdout);
f();