The scholarly suggestions above are fine if you want to puzzle out a Latin sentence or two, or if you have a need to demonstrate your cleverness and industry to a professor. If you want to read Latin, though, they are to be shunned.
Obviously, Romans didn't "hunt the verb" first, then "hunt the subject" and so on. They simply read a bit of writing start to finish, or listened to someone speak, and discerned the meaning naturally. This is what the student of Latin should strive to be able to do. It takes practice -- many words have several possible meanings, and so must be left ambiguous in the mind until the context makes them clear -- but it is the most profitable way to approach a Latin text.
In an 1887 speech called "The Art of Reading Latin: How to Teach It", William Gardner Hale expanded on this idea to great effect. I've lost most of the speech text, but following is a sentence analyzed by Hale. (I lifted the analysis and reworked it a bit.) The scene is from Livy, of two assassins who have just put an axe into Tarquin's head. Hale proceeds one word at a time, working out the context as he goes, as the Romans did subconciously and immediately.
Form? Proper noun, accusative singular.
Meaning? It's not the accusative of duration of time, clearly, because it's a man's name. For the same reason, we can rule out extent of space, extent of action, and cognate accusative. It's also not the accusative of specification ("as regards Tarquin"), which is very rarely used in prose, and there only with a few stock phrases. It's not an accusative of exclamation, either, probably; Livy was a historian, and not prone to call out to dead people. What, then? The direct object of a verb, an appositive to the direct object, or the subject or predicate of an infinitive. We can't know any more than that yet, so we'll just keep those possibilities in mind and keep reading.
Form? Adjective, nominative singular neuter, or accusative singular masculine or neuter.
Meaning? Latin word order is free, but some conventions exist; in particular, this word most probably belongs to "Tarquinium". In our minds, then, we have a picture of the dying Tarquin, acted or acting upon.
Tarquinium moribundum cum...
Form? Preposition or conjunction.
Meaning? There's no way to tell without more context. Important to remember, of course, is that not even a Roman could know the meaning of cum here without more to go on.
Tarquinium moribundum cum qui...
Form? Relative or interrogative pronoun.
Meaning? No way to tell yet here, either. The word does resolve the meaning of cum, though, which has to be a conjunction in this context.
Tarquinium moribundum cum qui circa...
Meaning? Modifies a verb, adjective, or another adverb. We'll make the decision when we can.
Tarquinium moribundum cum qui circa erant...
Form? Verb, third person plural imperfect active indicative.
Meaning? Ah, now we have some good information to go on. Qui must be a relative, because interrogatives require a subjunctive verb. Also, circa very probably modifies erant. What we have here, then, is a phrase ("qui circa erant") that we can treat as a substantive ("the bystanders", or something similar).
Tarquinium moribundum cum qui circa erant excepissent...
Form? Verb, third person plural pluperfect active subjunctive.
Meaning? Completes the cum-conjunction clause. The Romans were very fond of plucking out the most important word or phrase from a subordinate introductory sentence, too, so we can assume that Tarquinium is the direct object of excepissent. So: Tarquin is dying, and the bystanders have picked him up.
Tarquinium moribundum cum qui circa erant excepissent, illos...
Form? Demonstrative pronoun, third person plural masculine.
Meaning? Because the word is first in the clause, the people it refers to are of special prominence. Set as it is against the special position of Tarquinium in the preceding clause, and given the Roman propensity to juxtapose opposites, we can guess that the assassins are the referent. At all events, we know by its form that it can be a direct object, an appositive to a direct object, or the subject or predicate of an infinitive. Sound familiar?
Tarquinium moribundum cum qui circa erant excepissent, illos fugientes...
Form? Participle, third person plural present active, masculine or feminine, nominative or accusative.
Meaning? It very probably belongs to illos, because assassins would be apt to flee. Just in case, though, we'll allow in our minds that it might belong to some other word.
Tarquinium moribundum cum qui circa erant excepissent, illos fugientes lictores...
Form? Noun, nominative or accusative plural.
Meaning? The lictores were the king's bodyguard, so it's unlikely that they would be fleeing. For this reason, we'll guess that fugientes doesn't go with them, and that it in fact belongs to illos, as we first imagined. So illos fugientes and lictores are subject and object of a verb. (We can't know which is which, yet.)
Tarquinium moribundum cum qui circa erant excepissent, illos fugientes lictores comprehendunt.
Form? Verb, third person plural present active indicative.
Meaning? Livy was writing about past events, so this form must be a historical present. It's also the end of the sentence, and now we see everything click into place. The lictores have captured the assassins!
It's obvious that a fluent reader must have a thorough knowledge of Latin grammatical forms and syntax -- there can be a great number of possibilities to sift through for each word. Fluency takes practice, and lots of it. But the work does get easier, and eventually it becomes automatic -- just as English reading has become for all of us.
Note the vividness of the Roman word order, too: There's Tarquin. He's dying. The assassins have picked him up, and are running away. Now comes the king's bodyguard, and... they nab the bad guys!
How much more jejune a hunt-the-noun translation would have been! "When those who were nearby had picked up the dying Tarquin, the king's bodyguard captured them as they fled." Such stuff is too dull to withstand for long. It's no wonder so few people read Latin for pleasure any more.