First, let's talk cuticles. As you can see from the picture, the cuticle is actually the stuff at the base of the nail that grows up over your nail in a thin layer. It is NOT the nail fold of the eponychium (see the diagram)- which is what most people think it is. The cuticle that has grown over your nail can get in the way when you paint your nails, causing ridges and bumps.
Hangnails are the peeling skin at the base and sides of nails. The free edge is the part of the nail that is over the end of the finger.
I'm going to go over my whole nail care routine when I do a full manicure, and explain what I do and things I use that have really made my nails strong and healthy.
First comes filing. I file (and I recommend you file) with nail polish still on, it makes it much easier to see what you're doing and keep things straight. Also, I recommend you use a glass nail file like this (you can find them on Amazon), but be careful, they can really take down length quickly! You can also file in both directions with a glass nail file without making the edges ragged. Then I remove any existing nail polish. I personally use pure acetone. Acetone is very drying, but I feel that using pure acetone is much much faster at removing my nails polish than using non-acetone remover and really having to scrub at it to remove it, but an acetone-free nail polish remover is much healthier for your nails!
Once my polish is removed, I then do a 10 minute or so soak (one hand at a time) with a mani bomb. I use a nail soak bowl like this, with warm water, and I found some mani bombs on Amazon, though what I personally use is a crushed version of this.
I think actually they would be easy to make- it's fizzy like a bath bomb, and the ones I use have lots of good essential oils for moisturizing, which is the point of the soak. It also makes it much easier to push back and remove cuticles.
Then after my soak, I add a little cuticle remover (I use Sally Hansen's Instant Cuticle Remover) to the base of my nails and let it sit for less than a minute. Then I use a cuticle tool with a spoon on it to push back and scrape my nail beds (lightly!) to get rid of any cuticle that has grown over my nails. I also use cuticle nippers (I use Tweezerman cuticle nippers, they're super sharp) to clip my eponychium. This is controversial, you definitely need to be super careful and not cut too closely or you can cause damage. I just like the look of my nails better with it trimmed. You also must always cut any hangnails, never ever pull them. Not only does it hurt like crazy, but it can lead to infection. I actually carry a small cuticle nipper (well, a little nail kit with everything, lol) with me always!
Then after I've done one hand, I apply cuticle oil liberally to my nails. This is, along with the moisturizing mani bomb soak, is what counteracts the drying of the pure acetone. 786 Cosmetics has an Almond & Ginseng Cuticle Oil that would be great for this! Then I do the other hand and let the cuticle oil sink in for awhile. Oiling my nails is one of the things I started doing that I feel really has helped the health and strength of my nails. They're not nearly as brittle and peely now.