You're not yet ready for the task of accepting admin requests. The problem is that most requests will require only a few select skills out of the vast array of skills that you'll eventually learn by writing code, but most likely you won't have them or even know what they are. You are, however, ready to start writing your own code. Try making up a few small projects, like perhaps a simple calculator, a Visualforce page that shows this quarter's projected sales by line of business or some other grouping, and writing a database trigger that prevents duplicate records based on a couple of fields.
Try playing around with each project, intentionally trying to break it. What happens if you use "after insert" instead of "before insert" when trying to prevent duplicates? What things do you have to watch out for? What happens if you click the buttons too quickly on your calculator? Does it give incorrect results or miss input? How can you fix that? What happens if there's a negative value in some of your sales projections? Does the average still look right? Can you get enough code coverage to actually deploy all this stuff to production?
You need to know what you can do, what the platform can do, and how you can troubleshoot when things go wrong (and they will go wrong). You also need to know approximately how many things you can do, so experiment. Also, you need to learn how to unit test properly, so when things do break, you'll know why and how to fix it. You'll find that the more projects you do, the faster you'll get things done and the more productive you'll be. Once you've completed about a dozen projects satisfactorily, then consider asking your admin for some work.
Until you can consistently get high code coverage on your projects that includes a number of useful assertions, it's not advisable to take on too much admin work. Salesforce has some pretty strict limits on number of queries, memory, CPU usage, etc. You'll want to learn to be efficient at what you do along the way. Simply jumping right into admin work may have an adverse affect on the performance of the platform, and consequently, the satisfaction of the users.