Calligraphy has multiple meanings. Per m-w, the first two definitions:
So on one hand calligraphy means “pretty handwriting” and on the other it means “handwriting treated as art”. You’re trying to achieve the first; several of the people in this thread are referring to the second.
I’m left-handed and my handwriting improved markedly once I graduated from high school and no longer had to sit at a desk like this. Whoever designed these desks and convinced middle and high schools to use them exclusively is evil. Not only do they encourage the left-handed to hook their hand when they write, they also make you twist unnaturally in your seat to reach over to the writing surface, causing frequent back ache, as this person and this person also complains.
Now that I’m no longer made to sit at desks that make me contort myself to write, I write by holding my pen just the way a right-handed person does, but in mirror image. And people remark about how neat my writing is (at least when I’m not struggling to keep up with taking minutes!)
Put me in the cursive italic calligraphy column. Further suggestions boil down to:
Make yourself a lined template to place under unlined paper. Just print out a lined sheet using a word processing program. Use card stock, if possible. Then paper clip it to unlined paper. This will help you learn to write on a straight line and size your letters (especially ascenders and descenders) correctly.
Concentrate on the italic letter forms and the connectors. You really do need to practice writing individual letters. Pay special attention to the counters (blank spaces in letters).
Use a fountain pen with a medium italic nib. Cursive italic is characterized by the contrast between the broad and narrow strokes. This, in turn, will force you to hold the pen at a certain angle and form the letters correctly.
Take a calligraphy class. Many extension schools and community rec programs offer them.