"They" say that to become really, really good at something, like high-quality professional-level good, you need to work at it for at least 10,000 hours. I think this is true of writing.
Taking writing classes is great, but they have their limitations. I'm not an English major, but I've taken some university-level creative writing classes. Most of our writing exercises involve imitating other writers--which is good, you need to know what's worked in the past, and why it works. You develop technique. But at some point, you need to develop your own style too. I'm mostly a self-taught writer, and not a very good one at that, but I think that working on my own to hone my craft generally works far better than working in a classroom setting. But again, each person learns differently...
Writing is a solitary activity. Putting words together to form sentences is an art. Putting sentences together to form paragraphs, and paragraphs together to form stories is also an art. First, you have to understand the components of a sentence, the subject, the predicate, the indirect object pronoun. You need to know what you are writing, what it means, and why each word is necessary. By all means, break the rules later. But as they say, you gotta know the rules before you break them. Think about word choice. "Does this sentence mean what I want it to mean? Is this the right word? Does it fit in this sentence, or am I really reaching for another word that I can't find? Am I just settling because I don't want to go through the effort of seeking out the right word?"
When writing stories, try to keep an active voice. The passive voice is acceptable in certain situations, academic writing maybe, but it's dry and boring. "I built the house" vs. "The house was built by me." The first sentence is cleaner, more direct, and just better in most situations.
Of course, the most important thing you can do to improve your own writing is read. Read, read, read, read, read. Read writers you like and think about what they do well. Copy it. Read writers you don't like and pinpoint what grates you. Purge it from your own writing. Think about structure, tone, and cadence as you read. Read out loud, and feel the starts and stops, the pauses and breaths, the sounds of the words on your tongue. Pay attention to the details. Details make or break your writing.
The second most important thing you can do is to write. Write every day, a sentence or a page or ten pages. Just write. Observe your environment. Daydream. Don't aim for perfection right away, but don't be afraid to part with bad writing if it's just...bad. I have a tendency to cling to terrible writing just because it's mine, and I put time into it, and it has "potential." That's the wrong attitude. Separate the wheat from the chaff, figure out what you're doing wrong and what you're doing right. Keep the good, chuck the bad. But don't get angry at yourself if it's not going well, as you should be learning something from every writing experience, good or bad.
Read Strunk & White. Emulate writers you admire and draw from them until you develop your own ear and voice. Make your prose like poetry--that is, try to convey that which is impossible to convey. Use metaphor and descriptors (adjectives and adverbs) in small but effective doses.
Finally edit, edit, edit. Proofread. Add and remove. Does the story make sense? Are the characters developed? Why does X do Y? Why do I need this sentence here? Can I explain this process more effectively? How do I make this chapter more interesting? Have others read your work. Ask for feedback. Critique others' writing, and it will teach you to critique your own. An editor's work is never done.