Well, there are probably a dozen ways to go about this. For example, you can download Moodle and install it on your PC or Mac. I have had moderate success doing this. For example, I can't get Moodle 1.9.9 to run in my Windows 7 environment, but I can get 2.0 to run in a Windows XP. Neither of these are high priorities, so I don't bother much trying to fix Windows 7. Also, with this method, I am not sure if you could simulate your webpage idea. But this might satisfy your "local machine" idea.
Going live is not really going live because you always have the ability to uninstall or turn off the system. I am experimenting with Moodle 2.0 right now. My install is on the Internet, but no one knows about it. I locked out Guest login, so no one except I can get in. I plan to remove this at some time in the near future after I learn what I need to do for my "live" install (the one that students will access).
There are many alternatives to running Moodle on a hosted server. The startup cost is less than $30 US for about three months (including a new domain), and then you would be looking at around $5-$7 per month. This works, I did it for 4 1/2 years, but it is not the most powerful environment. Well, for your 85 students, or for experimenting, it might be fine. This solution would give you the website functionality that I think you are seeking. However, I am not sure if you could run version 2.0 or higher. The last time I did this on my hosted server, only 1.9.x was supported.
If you have your own server, installing Moodle shouldn't be too hard for you. I use (actually, pay for) a virtual dedicated server. I think this kind of solution would work for 85 students, but it is more involved. I am not sure how much technical knowledge you have.
Another solution is the company Moodlerooms. I am not sure how expensive they are, but people who use Moodlerooms seem to be happy. I am not sure if they could support your website needs.
If you want a rock-solid installation, you might need to hire a person with Apache, PHP, and MySQL experience. I have been really impressed by some of these folks (I am a novice).
If you are looking for ideas, you can always visit my website (www.rjerz.com). You will see how I have bundled Moodle to be inside my website. This is opposite of what Mary suggested. If needed, I can let you log into one of my courses and see how I do things. My solution is only one of many. Mary's idea looks like a very interesting approach. For me, I use Dreamweaver for my website and content management, and then link to the components within Moodle. There are pros and cons of this approach, but it works for me.
The solution also depends upon your budget. From what you have shared, time might be much more important.