Recall that we can split words up into syllables, e.g. English "pronounciation" = "pro-nun-ci-a-tion". As is described in that Wikipedia article, an "open syllable" is a syllable which ends in a vowel and a closed syllable is one that ends in a consonant, e.g. in "pronunciation" the open syllables are "pro", "ci" and "a", while the closed syllables are "nun" and "tion".
Also recall that in English and Hebrew words have what is called stress on one of the syllables. For example, the English word "contest" in the sentence "I won the contest" has the stress on the first syllable (CON-test), while the word "contest" in the sentence "I want to contest that ruling" has the stress on the second syllable (con-TEST). In Hebrew the stress is most often on the last syllable (e.g. מָכַר ma-KHAR), but is sometimes on the second-to-last syllable (e.g. מֶכֶר ME-KHER). In the Tanakh the ta'ame hamiqra are placed on the stressed syllable, but in general one must just know which syllable is stressed by being familiar with Hebrew. A few texts mark the stress with a meteg (e.g. the Artscroll siddur when the stress is not on the last syllable), or with an ole.
Qamas Qatan and Gadol
Now most Sephardim have two ways to pronounce the qamas vowel. It is sometimes pronounced a (e.g. חָפַר) and sometimes o (e.g. חָפְשִׁי). When it is pronounced a it is called a qamas gadol, and when it is pronounced o it is called a qamas qatan. The general rule is that a qamas is always a qamas gadol unless it is in an unstressed, closed syllable.
- חָפְשִׁי = hof-SHI: The syllable khof is unstressed and closed and therefore has a qamas qatan.
- חָפַר = ha-FAR: The syllable kha is unstressed, but it is not closed (it is open), so it has a qamas gadol.
- עוֹלָם = ʿo-LAM: The syllable LAM is stressed, so it has a qamas gadol.
Of course there are a few complications, and I'd like to mention two specifically. Firstly, one must remember that a consonant with dagesh hazaq (i.e. dagesh on a letter other than one of the letters בגדכפ"ת) is pronounced as doubly long, and this affects syllabification. For example, in חָנֵּנִי = hon-NE-ni the first nun is doubled, so the first syllable is closed and unstressed, and therefore has a qamas qatan. Secondly, the word כָּל is almost always pronounced as kol.
Shewa Naʿ and Nah
The details of exactly when a shewa is naʿ (i.e. pronounced as e) and when it is nah (i.e. silent) are complicated, but here are the basic rules. There are five rules that you can remember by the siman אבגד"ה:
(א) A shewa is always naʿ at the beginning (א׳) of a word. Example: קְרִיאָה = qeria
(ב) A shewa following two (ב׳) consecutive consonants is always naʿ. In other words, when two consonants are written next to each other and both have a shewa, the second shewa is naʿ. Example: אֶשְׁמְרָה = eshmera (Both שְׁ and מְ have a shewa and are next to each other.)
(ג) A shewa under a letter that follows a tenuʿa gedola (ג׳ for גדולה), a long vowel is always naʿ. The following vowels are the tenuʿot gedolot: אָ when it is qamas gadol, אֵ ,אִי ,אֹ ,אוֹ, and אוּ. (Note: this does not include אִ, אָ when it is qamas qatan, or אֻ). Example: שוֹמְרִים = shomerim.
(ד) A shewa under a letter with dagesh (ד׳) hazaq is always naʿ. Example: אֲכַפְּרָה = akhappera.
(ה) A shewa is always naʿ between two identical (ה׳ for הַדּוֹמִים) consonants written with separate letters (rather than dagesh hazaq). Example: וּלְלֵאָה = ulelea.
The most complicated rule is definitely ה׳, and there are some seeming exceptions, but אין כאן מקום להאריך. I do think it's worth noticing when qamas is followed by sheva in an unstressed syllable, as in the word שָמְרָה, there are two possibilities -- either the first qamas is qamas gadol and the shewa is naʿ (so shamera), or the first qamas is qamas qatan and the shewa is nah (so shomra). In fact, these are both words with different meanings, so with a basic siddur you would need to use context to figure out which is correct each time it shows up.
For more information
I believe the Rada"q discusses all of this in his Sefer HaMichlol. Tiqqunim also often have a section in the back about diqduq that covers these topics and more.