No matter if you're self-publishing or looking to publish with a press, if you can't use grammar correctly, you're not going to get very far. A couple typos or misused punctuation can ruin your chances with agents, publishers, and readers.
Semicolons are one of those punctuation marks that most people don't know how to use, so in order to help you, I'm going to give you a quick rundown on how these weird things work.
Semicolons have 3 uses:
separate two closely-related complete sentences
clarify lengthy, obnoxious sentences
clarify post-colon lists
Okay, so this sounds really scary, but let's do some fun examples. I'm even going to use Harry Potter examples, so get pumped.
Separate Two Closely-Related Complete Sentences
Harry Potter is the Boy Who Lived; he plays Quidditch.
So that above? That is not the proper use of a semicolon. While there's a complete sentence on either side of the semicolon, the subject matter is not closely-related enough to use a semicolon. So let's change it.
Harry Potter is the Boy Who Lived; he lived in a cupboard under the stairs.
This is much more appropriate. There's a kind of mirroring in the sentences, and that makes it a much more appropriate use.
Clarify Lengthy, Obnoxious Sentences
Harry Potter, the Boy Who Lived, lived beneath the stairs while his abusive aunt and uncle lived in relative luxury, Hermione, girl genius, lived with her supportive Muggle parents, and Ron Weasley, poor and Pureblood, lived in the Wizarding World with his loving parents and half-dozen siblings, as well as a gnome-infested garden.
That's hard to look at, isn't it? And even harder to read. When you have long complex sentences like this, you have two options: use semicolons or make separate sentences.
If you decide to use a semicolon, here's what it would look like:
Harry Potter, the Boy Who Lived, lived beneath the stairs while his abusive aunt and uncle lived in relative luxury; Hermione, girl genius, lived with her supportive Muggle parents; and Ron Weasley, poor and Pureblood, lived in the Wizarding World with his loving parents and half-dozen siblings, as well as a gnome-infested garden.
This is much easier to read. The semicolons act like miniature barriers to show where the separate pieces of information exist. However, it's probably best to avoid this in your writing. Readers are turned off by very long sentences such as the above, and if it's difficult to read or understand, then they're going to be pulled out of your story.
Clarify Post-Colon Lists
This is very similar to the one above, but let's look at an example:
There are three main characters in Harry Potter: Harry Potter, the Boy Who Lived, Hermione Granger, girl genius, and Ron Weasley, poor and Pureblood.
Like the sentence from the last section, this isn't very clear, is it? Some semicolons will make that better, though:
There are three main characters in Harry Potter: Harry Potter, the Boy Who Lived; Hermione Granger, girl genius; and Ron Weasley, poor and Pureblood.
This clarifies which details go with which character and makes it easier to read. You probably won't use this very often in creative writing, but it's still good to know.
Last Tip on Semicolons: Use them sparingly. A semicolon is not as commonplace as a period or a comma. Its appearance is a signal to readers that what you're about to say is very important. Reserve semicolons for these important comparisons and try not to use more than a handful.