How do I free up space on my Mac?
First step would be to go through your files and get rid of things you no longer need.  An app like GrandPerspective can make it easy to locate files that are taking up a lot of space.

If you’ve cleaned your drive of all unwanted files and you still don’t have enough space, then you should consider replacing your internal drive with a larger capacity drive.  Just make sure to get one big enough that you won’t fill it up for quite a while.

If you don’t want to replace your internal drive, then the next best option is an external drive.  As for the caveats you mention, you could just leave the drive mounted all the time.  That way the files are always available to you when you need them.  And for keeping track of files that need to be backed up, just get a good backup app, like ChronoSync.  They are designed to keep track of modified files and back them up on whatever schedule you designate.