The most important tip I can give you to make your writing neater, as with improving any skill, is to practice. Nowadays it's so easy to write down almost everything on a phone or computer. To practice, carry a pad of paper and a pen with you and manually write down any lists, reminders, or memos instead of typing them into your phone.
It's also easier to write neatly by using a thicker tip pen (try a 0.5 tip). Using thin ballpoint pens can make your writing look scratchy and less neat.
If you have some free time, practice writing out this sentence: "The quick brown fox jumps over the lazy dog." This sentence contains all the letters of the alphabet and will help you practice all your letters, but it won't seem as repetitive as simply writing out the alphabet. You could alternate between lower case letters, upper case letters, and even cursive if you are also learning cursive.
Writing and calligraphy are dying arts, but nowadays writing is still very necessary in school. Keep all your progress in a notebook so you can go back through the first pages and see how much you've improved! Good luck!