If you're willing to put some time into configuring software, there are a handful of free Learning Management System (LMS) platforms out there. The best example is Moodle, a FOSS platform. I've always been a fan of something called Open edX because it's got some neat social aspects. An advantage of a real LMS platform is that you can drop in any online training courses that you build/buy later on.
I don't have a sample to share, but before implementing a corporate LMS we managed a very simple database that tracked a unique employee ID, a course ID#, and a session ID (a session = 1 offering of a "course"). If you're going to keep it simple and focus on simply tracking activity after the fact, I recommend taking the time to build a database and not live in spreadsheets. That's just my experience.
Well, Moodle is a free tool. It's geared for education environments rather than corporate. And it would require some nontrivial setup and maintenance on your end. But it fits your price point.
Definitely don't discount the administrative work. But if you're only using it for reminders and quizzing it might be better than your manual processes.