What do you like best?
Moodle is highly customizable allowing you to completely redesign how your course looks, how it is organized, and what tools are fully integrated into the LMS. There are a ton of free themes available for various age groups or purposes that give you a lot of flexibility as to how you want to organize (by topic, week, unit, etc.) and present your site (for K-12, university, public online courses, etc.) . There are also a ton of tools that are fully-integrated (not LTI like other LMSs) so that the students never have to leave your site to complete their work or lessons. There are fully integrated games and other learning experiences that make it easy for students to review vocabulary terms or practice skills. Within most activities/tools, you have almost complete control with a lot of options and settings to customize your delivery or how you want to integrate the activity into your course.
What do you dislike?
Since integrating certain types of assets requires SSH connectivity to the server, I've found that it was easier to simply maintain my own moodle server, but this requires a tremendous amount of knowledge and skill to maintain a secure server. This is not something that someone without a strong computer security background should attempt. While there are tools (like bitnami) that make it easier to get a Moodle site up and running, some of the customization and more interesting plugins require a little back-end knowledge or access to get working right. However, if you're just looking for relatively standard features (lessons, files, progress bars, games, etc.) - that's all easy to install onto your site and easy to configure. I have a few issues with specific plug-ins where they are not as flexible or don't follow the "standard" Moodle way of doing things, but that's not necessarily Moodle's fault. However, I feel like it is Moodle's responsibility to provide (and enforce) a common set of tools. For example, Moodle has the ability to support drag-n-drop reordering of widgets/items on the site and yet not all elements of Moodle take advantage of this. Like the gradebook, for example, I can't easily reorder items or move them into other sections (additional dialogs pop up, and then you have to scroll through dozens of items to find the right place) - when simply drag-n-dropping them should move them to a new order or to a new category without any additional issues. Of course, I've not seen other LMSs with better tools, so I don't fault them too much for this!
Recommendations to others considering the product:
If your school is using Moodle, then it is a great, easy tool. The flexibility and plug-ins will satisfy just about any need you have and the fact that they're fully integrated into Moodle, your students will find it seamless to use and intuitive to navigate. It's great for organizing and posting content - everything is on the top-level and immediately accessible, nothing is hidden from the faculty member or doubly linked or in multiple places. Take a few moments to explore the plugins library - it is an amazing set of tools that provide an enormous variety of ways for students to get involved with the content!
If you're building and administering your own server, Moodle can be quite an undertaking (not that any other LMS would be easier). Definitely use a tool like Bitnami to help you install and manage it or contract through an ISP that will support Moodle for you - this will solve 95% of your administration issues with Moodle and make maintenance much easier. But, for the flexibility of being able to install anything you want/need and configure the site to your heart's desire, using Moodle can be amazing!
What problems are you solving with the product? What benefits have you realized?
I teach programming at the university level and the most important feature for me in Moodle is the Virtual Programming Labs - they allow me to create short or long auto-graded programming assignments that are fully integrated into the Moodle platform. My students have truly benefited from these experiences as they get immediate feedback on whether their code is correct or not. There are some limitations to this plug-in, but it has overall been a fantastic tool that has taken a LOT of grading and evaluation off of my plate.
Students have commented on how they like the organization of Moodle over the other LMS my University uses and I certainly like its transparency. I like how easy it is to post something to Moodle and see that it is "live" or "hidden" from students - everything is right there, at the surface level, nothing is hidden or linked in some odd way. I am able to easily show my students their progress with check-offs next to each item and overall progress bars. My students are game development students, so they like being able to "complete" objectives.