I'm not going to re-hash what is already out there in the way of really good instructions on this, but I will give you my two cents on tips......
I would get some cheap disposable rubber gloves, your hands will thank you. Remember, this is nail polish, so it's going to stick to your fingers and nails if you don't (I had to scrub using finger nail polish to clean my hands because I didn't wear gloves).
I got my plastic container (for dunking) and the mugs at the Dollar Store, the nail polish I bought at Walmart...again, keep the cost down since they look like a million bucks when they're finished!
I didn't care for my second or third attempt....so I removed the colors using nail polish remover and a little elbow grease and went at it again.
Remember, these are not dishwasher safe....hand wash only. So be sure if you are gifting them to let the recipient know how to care for them. Also, try to leave blank space near the top of the mug for "lip room" just for good measure.
These could be really fun favors for baby showers (or bridal showers), use nail polish colors to match the gender of the baby, fill them with some candies or other goodies wrapped in tulle and you have a unique gift for each guest at a very reasonable price.
Note: If someone were squeamish about using this as a coffee mug with nail polish on it, then suggest they use it in an office as a fun pencil/pen container, or use it as a planter, or in the bathroom for a toothbrush holder, etc.