Starter CLAUSES often begin with when, while, after, since, because, although, as, if, etc. You can learn more about subordinating conjunctions here.
After the Trojans brought the wooden horse into the city, the Greek soldiers inside the horse waited until nightfall to begin their attack.
Because they caught the Trojans by surprise, the Greeks were able to capture the city in a single night.
If the Trojans had been more cautious, they might have been able to save themselves and their city.
The Greeks having overrun the city, there was nothing that the Trojans could do.
For more about the Trojan Horse and the fall of Troy, see Wikipedia.
Starter PHRASES include participial phrases, infinitive phrases, absolute phrases, nonessential appositive phrases, and long prepositional phrases (especially prepositional phrases that are four words or longer).
Grabbing hold of the sacred Palladium, Cassandra prayed to the goddess to save her.
Grabbing hold of the sacred Palladium, Cassandra prayed to the goddess to save her.
The Greeks having overrun the city, there was nothing that the Trojans could do.
During that single night of terror, the Greeks won a war that had lasted for many years.
Starter WORDS usually provide a kind of transition from the previous sentence. Common starter words include however, nevertheless, yes, etc. Sometimes there is a starter adverb that qualifies the whole sentence: fortunately, surprisingly, etc. (as opposed to simply modifying a verb).
The Trojans had put their trust in the gods, in their human allies, and in their own strength. Nevertheless, they were defeated by the Greeks.
The Trojans had put their trust in the gods, in their human allies, and in their own strength. Nevertheless, they were defeated by the Greeks.
The Greeks finally won the war. Yes, it was a great victory, but it came at a terrible price.
The Trojan hero Aeneas fled the fallen city. Fortunately, he found his way to Italy and made a new home for himself in that land.
For more about the Trojan Horse and the fall of Troy, see Wikipedia.