How to clean my eye glasses really well?

Get a cleaning kit. Get a cleaning kit specifically for glasses at your local drug store or walmart. It is inexpensive and well worth it.
Two ways. Microfiber clothes work well. In our professional optical shop, we use alcohol-based eyeglass cleaner and lens paper. These are available on the internet and at most optical shops. For really dirty glasses, we use an ultrasonic cleaning machine. A local optical shop probably has one of these and will clean them for a small fee.