Five conventional ways of putting sentences together.

View these as options, options to give your sentences variety and rhythm.
(Note that the highlighted rules are examples.)

1. The most common are conjunctions: and, but, or, so, (nor, yet, for) (fanboys).
Rule:
Usually two independent clauses (in effect, sentences) are joined by a conjunction, but a comma must be placed before the conjunction.

2. The most useful are adverb clauses (although, when, because, since, until, if etc.). These words convert an entire sentence into an adverb. For example: When these words convert an entire sentence into an adverb . . .
Rule:
When you use an initial adverb clause, place a comma after it.
Do not use a comma when you use a final adverb clause.

Notice the rule also illustrates that adverb clauses can move around in a sentence. Adverbs are really the only parts of speech that are easy to shuffle in this way.

3. The hardest to remember how to punctuate is sentence connectors: (however, moreover, furthermore, as a matter of fact, additionally etc.) The two-dollar term for these is adverbial conjunctions, so they can also move around in sentences.
Rule:
Don't hesitate to use sentence connectors; however, remember to surround the connector with punctuation.
Don't hesitate to use sentence connector; remember, however, to surround the connector with punctuation.
Don't hesitate to use sentence connectors; remember to surround the connector with punctuation, however.

4. The shortest is the semicolon: ;
Rule:
Use semicolons sparingly; use them to connect sentences connected in meaning.
(And remember that a semicolon is not the one used to introduce a list. The colon (:) is.

5. The powerful one, the verbal. Think about employing some of the thousands of verbs in sentence combining.

Quasi-rule:
Poised at the beginning of a sentence, the verbal construction was followed by a comma.

Or an example closer to home: The paper is graded by a computer. The paper is then ready for student revision.

Could become: Graded by a computer, the paper is then ready for student revision.

Be careful not to mislead or confuse the reader; check to make sure there is a referent for the verbal construction in the sentence (italicized in the sentence above). And contrast that sentence with the one below where the reference may be ambiguous:

Graded by a computer, the teacher can then return the papers.