Ok, on to my tools and tips!
This is the nail art brush I use for almost every nail art design I make. As you can see, the brush bristles are very short, and there aren't a lot of them. You do not have to use this exact brush, but something this small will allow you to get very fine detail. It is also important to thin your paint down with water so that you can do small details. A big glob of thick paint will not make the fine details you are hoping for.
Here's a closeup of the nail art brush I use most often. I purchased this at Sally's Beauty Supply for about 5 dollars. You can get your brush anywhere, it doesn't have to be a brush made specifically for nail art. It does have to be small. Robin Moses has a great video on how to cut brushes to make them work for you. Watch it and learn!
I paint my nails first using regular nail polish (it can be any finish), and then I use acrylic paint to do the nail art. These are some of the bottles of paint that I use. I picked these up at Michael's craft store. They cost about $1 each, but they will sometimes be on sale for even less. These bottles of paint will last a long time. I have yet to need a new bottle of black or white paint (which I use with nearly every manicure) and I have done at least 50 manis with these so far. This is not something that will be really expensive to add to your nail art supplies. I have the basic colors (red, orange, yellow, green, blue, purple, and brown) plus black, white, and 'flesh tone' peach. I also purchased bottles of gold, silver, and bronze acrylic paint for metallic accents. You can easily mix any of these colors together for your artwork.
This is the palette I use. It is a plastic lid. If you buy things like tubs of yogurt or deli items with lids, just save the lids and wash them off. I usually will use a lid about 5 times for different manicures before I toss it out. I also use a small plastic cup (it is pretty shallow) filled with water to wash out my brush between colors. I keep a folded paper towel or napkin next to my work space to wipe my brush on.
Please let me know if you have other questions about my technique or tools, and I will do my best to answer them! Thanks for reading!