asyncio is just nice API for async programing and event loop management. You should start from something high level in order to obtain asyncio ideas, like scraping [1] or try web development with aiohttp [2]. And then if you want to write some library or port database driver, look for examples at aio-libs[3] github organization.
[1] http://compiletoi.net/fast-scraping-in-python-with-asyncio.h...
[2] https://github.com/KeepSafe/aiohttp
[3] https://github.com/aio-libs