Summarize:

Some people prefer pen, others pencil.  Some like them larger, others smaller.  The important thing is to find a writing instrument that feels right in your hand.  Consider a pen or pencil with a soft grip, especially if you tend to grip too tightly. Use lined paper for practice, and sturdier paper if you are writing something to keep. Yes, your mother was right -- posture counts.  Hunching over your paper will make your neck and back sore after a while, and it also restricts your arm movement so that you will over-utilize your hand and wrist in writing (see Method 2, Step 3 below). If you can sit up straight as a rail and be comfortable, great.  But don't make yourself overly rigid and uncomfortable.  Beautiful writing should not be a painful chore. Grasp the pen, don't choke it.  (They say a good craftsman never blames his tools.)  If you have an indentation or red mark on your finger when you finish writing, you're holding on too tight.  A looser hold enables a better range of motion and allows the letters to flow more freely from your pen.  There are plenty of “right” ways to hold a pen or pencil.  Some pin it against the middle finger with the index finger and thumb, some press with the fingertips of all three fingers; some rest the back of the pen on the index finger base knuckle, some on the webbing between index finger and thumb. Instead of spending time forcing yourself to utilize a new grip, go with what is comfortable for you -- unless you find that you use an awkward grip that negatively affects the quality of your writing.  So long as you're utilizing your first two fingers and thumb, it should work fine. Sure, it's fine to use abbreviations, symbols, non-sentences, etc. when jotting down notes, but especially when it's something that another person will see, take the time to write properly.  A shiny, clean car that's missing two wheels and the hood doesn't look as beautiful as a complete one.  Make sure you have the appropriate capitalization and punctuation marks. Don't use text speech or internet abbreviations. If you are writing anything other people are going to read, don't use text writing: Gr8, bcuz, u, soz, lols, etc. Do you know someone with beautiful handwriting?  Watch him or her write and ask for some pointers.  You may even want to look at word processor fonts for inspiration on letter shapes. Don't be too proud to seek out writing lessons and workbooks marketed for schoolchildren.  For that matter, if you have children, practice together.  Turn family bonding time into better penmanship for all.
Choose the right tools. Sit upright but comfortably. Hold on loosely. Make your content more beautiful. Seek inspiration.