Independently of Michael, I was going to post a random Xenophon passage with the participles highlighted. Here it is. Red is for participles, blue is for infinitives, and green is for finite verbs.
Xenophon, Anabasis 3.2.11-12
ἐλθόντων μὲν γὰρ Περσῶν καὶ τῶν σὺν αὐτοῖς παμπληθεῖ στόλῳ ὡς ἀφανιούντων τὰς Ἀθήνας, ὑποστῆναι αὐτοὶ Ἀθηναῖοι τολμήσαντες ἐνίκησαν αὐτούς. καὶ εὐξάμενοι τῇ Ἀρτέμιδι ὁπόσους κατακάνοιεν τῶν πολεμίων τοσαύτας χιμαίρας καταθύσειν τῇ θεῷ, ἐπεὶ οὐκ εἶχον ἱκανὰς εὑρεῖν, ἔδοξεν αὐτοῖς κατ᾿ ἐνιαυτὸν πεντακοσίας θύειν, καὶ ἔτι νῦν ἀποθύουσιν.
Brownson/Dillery tr.:
For when the Persians and their followers came with a vast array to blot Athens out of existence, the Athenians dared, unaided, to withstand them, and won the victory. And while they had vowed to Artemis that for every man they might slay of the enemy they would sacrifice a goat to the goddess, they were unable to find goats enough; so they resolved to offer five hundred every year, and this sacrifice they are paying even to this day.
Here’s Watson’s “literal” translation, which might give you a better idea of the grammar (or not):
For when the Persians, and those united with them, came with a numerous host, as if to sweep Athens from the face of the earth, the Athenians, by daring to oppose them, gave them a defeat; and having made a vow to Diana, that whatever number they should kill of the enemy, they would sacrifice to her divinity the same number of goats, and not being able to find enough, they resolved to sacrifice five hundred every year; and to this day they still continue to sacrifice them.
To an extremely rough and inaccurate first approximation, Greek participles are “-ing” words in English (gerunds). Greek infinitives are English infinitives, “to …” formations in English. Any noun in a Greek sentence, (ie. a subject, direct object, indirect object), can be a noun phrase with a participle doing something.
The girl pushed him → The girl running by pushed him
“Running” (note the gerund) would be a participle in Greek.
And, again to a first approximation, infinitives represent conceived action. In English you “try to do something” or are “able to do something”, and both of these constructions take the infinitive in Greek as well. In general, if it’s a construction that has to be a verb and couldn’t be replaced by a simple noun, it takes the infinitive. But there are major and frequent exceptions, which even occur in the above Xenophon passage.
I realize that there is some disinformation in the above, but it will hopefully be enough to get you started. At some point once you have read enough, participles (which are more frequent and useful in Greek than gerunds are English) begin to make to sense.