Just so I don't have to write it out in every item here, there are two things I can suggest to catch and/or fix these problems. First, acquire a spellchecker program that also looks at grammar. Microsoft Word has a pretty good one built in, but I'm sure there are cheaper or possibly free ones out there as well. Just make sure to actually pay attention to what they're telling you is wrong and how they're fixing it rather than blindly clicking through the prompts to fix errors, that way you can learn from the mistakes and avoid them in the figure. The other way to go about it is to read your writing out loud to catch problems. If you speak English on a regular basis, you're probably better at catching verbal mistakes than you are at catching written ones. This is because you likely speak the language far more than you write it, plus there are some really nice verbal cues (like pauses for commas and periods) that help you catch when people are saying things strangely. Just read your work aloud and fix anything that seems off, it's as simple as that. You might be surprised with just how many mistakes you can catch with such a simple thing.
Tenses: There are two tenses commonly used in fiction writing: past tense and present tense. Past tense makes use of the past tense form of verbs, such as ran, jumped, kicked, slid, and sat. Present tense makes use of the present tense form of verbs, such as run, jump, kick, slide, and sit. For the most part, you'll want to keep your verbs in the same tense throughout your writing. For example: "Amy beat up some dude in an alleyway and took his money. She ran away when she heard police sirens, kicking the chump again for good measure on the way past. Amy gets in her car and drives away." Note the difference between the underlined past tense verbs and bolded present tense verbs. This shouldn't be happening in your writing because it is confusing and conflicting.
Note that there are a couple exceptions to this rule, the main one being dialogue. Whenever someone speaks it should be in the present tense, unless they're talking about events in the past from their perspective. This can be sort of confusing, but think of it like this: reading present tense writing is like watching events happen in real life; reading past tense writing is like watching a recording of events that happened at some point in the past. If someone says "I hate bikes" in real time, they wouldn't magically change to "I hated bikes" on a recording of those words just because it happened in the past. The same thing goes for the opposite sort of thing: somone saying "I crashed my car" on a recording wouldn't say "I crash my car" to your face when telling you about the accident they got into yesterday (unless they're giving you a sort of play-by-play of events from their perspective). Just be sure your verbs associated with the dialogue, such as says/said, match the tense of your non-dialogue verbs. The other exception to this is verb forms that end in -ing, which are called gerunds. When used appropriately, these types of verbs don't mess with the tense of your sentences; see the word 'kicking' in the example in the previous paragraph to see this at work.
Sentence Fragments: A sentence fragment is anything that is written like a full sentence, with periods to separate it from other things, but does not constitute a full sentence by itself. This may be because it lacks a subject or a verb or it's a subordinate clause that needs to be attached to an independent clause to make sense. Fragments can be spotted pretty easily by reading a sentence without looking at anything around it to remove any context that could mask the problems. If the sentence doesn't seem to make any sense, doesn't give a complete thought by itself, it's probably a sentence fragment.
For example: "The students working in the lab last semester. Performed experiments with sulfur." Both of these are sentence fragments. What about the students, what were they doing? Who exacly performed experiments? They should be together in one sentence, separating them is grammatically incorrect. Not all sentence fragments will be that easy to spot, nor will they all be corrected in that same manner, but it makes for a fair example.
Run-on Sentences: Despite the misleading name and the resultant common misconception, run-on sentences have very little to do with the length of a sentence. A run-on sentence is a sentence in which two or more independent clauses are joined without proper punctuation or conjunctions. For example: "I ran to the store I bought ice cream." This is a run-on sentence because there's no separation between the two independent clauses. These can be fixed in a variety of ways: turn them into two sentences ("I ran to the store. I bought ice cream."), use a semicolon("I ran to the store; I bought ice cream."), use a comma and conjunction to separate them ("I ran to the store, and I bought ice cream."), or restructure the ideas so they're no longer both independent clauses ("I ran to the store and bought ice cream." or "I ran to the store to buy ice cream.") Which fix you'll want to use really depends on the exact nature of the run-on sentence, just try to pick whatever seems to fit best and you should be alright.
Comma Splices: These are kind of a form of run-on sentence. Comma splices are places where two independent clauses are separated by just a comma. If you're careful to watch for run-on sentences, you should be able to spot or avoid these just fine. Note that comma splices are sometimes okay for stylistic reasons, especially in poetry and dialogue. The famous sentence "I came, I saw, I conquered" is technically erroneous because of the comma splices, but it flows so much better than alternative ways the sentence could be restructured. You can get away with some splices, but use them sparingly and only where they really make a positive difference in the flow of your writing.