How to stream the output of a command in a bash script?

Adding a command in a script will simply print the command output to the console.
But what if I want to filter the output, add to it, act upon it, etc.?

Right now, I simply assign the output of the command to a variable, print the variable and grep the variable to find the bits that interest me.
But if the command takes a long time to run, the user won't see any output until the command has finished running.
I want to do all of the above, while the user is seeing the output and the process progresses...