Is it appropriate to email them and let them know I am taking these steps
Yes, that's appropriate. It sounds like the contact you had was leaving the door open to future contact. That said, I would keep a few things in mind:
- Be specific in your replies. "Hey, you got any jobs?" may just come across as annoying. If you can reference specific topics you talked about, that's better.
- Be ready to show some progress. You mentioned you were perceived as short on a certain skill set but you're taking classes. Mention that.
- Be ready to discuss what you've been doing in the meantime. If it was JUST taking the classes, because they were full time, that's great. But were you also working? If you simply come across as if you've been sitting around doing nothing for several months, that won't reflect well on you.
Keep the email as short and to the point as possible. Busy people don't like being faced with an unexpected wall of text. If you can manage those three points in one or two sentences, that's great. Three or four paragraphs will probably just get a delete.
Also, in a more general sense, it's best to consider this reach-out as part of your larger strategy of targeting this employer, versus this reach-out being your ONLY contact with them. You hinted at this here:
I am also trying to find more, non annoying ways to stay on their radar. Any ideas?
This will be specific to the employer. The key here is to learn about their hiring channels and use them. Do they have a recruiting page? Keep an eye on it. Do they go to recruiting events at the school where you're taking classes? Stop by and talk to them. Do they have a linkedin profile for the company? Follow it. Is your contact there on linkedin? Add them as a connection. Linkedin (or other professional networking sites) can be a great way to "stay on the radar" without being annoying. Connect with the people who you know at the company, and then make posts that show what you're learning and what you're accomplishing.
I really like seeing a candidate who is making relevant posts and talking about their progress or skills development, but you need to make sure it's relevant and doesn't come across as self-centered or braggy. It's fine to talk about projects you're doing in your classes, and if you see articles or blog posts you learned something from, link to them and post a few sentences describing why you liked them.
Finally, keep your eyes open. It's great to have a dream job and a dream employer, but there are probably lots of other great jobs out there. Going after other opportunities, while staying in touch with your dream employer, can be a great way to contribute to an organization, develop your skills, and maybe find a different dream worth chasing - versus being fixated on a theoretical "dream job" that may never materialize.