The Cyrillic alphabet, though most notably used by Russian, is also used by literally dozens of other Slavic languages, from Serbian to Silesian to Slovene to Old Church Slavonic. Though pronunciation rules stay largely the same, if you are planning on learning other Slavic languages, then it would be best to find language-specific pronunciation rules.
It should also be noted that this is all typescript Cyrillic: handwritten and Cursive Cyrillic differs greatly. However, in this modern age, with less and less being written by hand, this discrepancy is becoming less of an issue.
Typing the Cyrillic alphabet on a Latin keyboard is another small hurdle, but one easily managed. To type in Russian, check out your operating system's method of changing keyboard layouts, and then make or purchase stickers to put on the keyboard.
One of the major distinctions to be made about pronouncing Russian words is palatalization. One may think of palatalization like the softening of the sound by blending and blurring two letters together. To get a feel for it, try listening to recordings of spoken Russian; written explanations can't really do it justice.
Without further ado, the Cyrillic alphabet for Russian:
А, а *: Not only does it look exactly like the Latin alphabet A, both of them being derived from the Greek letter alpha (Α, α), but they also share much the same sound. Consider the “ah” sound in “father”.
Б, б: Pronounced “be” in almost exactly the same manner as the Latin B. When placed at the end of a word, it may also take on a more subtle, almost voiceless “p” sound. Try not to confuse it with the number 6.
В, в: Here is where the Cyrillic alphabet begins to get a little confusing. While this looks precisely like the Latin B, it is pronounced like the Latin V, “ve”. Both this and Б are derived from the Greek letter beta (Β, β), which used to represent both the V and B sounds.
Г, г: This letter, derived from the Greek letter gamma (Γ, γ), is a little difficult to pronounce. For English speakers, it sounds like something between a hard G sound and a K. At the end of a word, it makes a subtle K sound. Rarely in certain declinations, it may even make a V sound.
Д, д: This letter, derived from the commonly used Greek letter delta (Δ, δ), makes the Latin D sound that we are familiar with. When found at the end of a word, it makes more of a slight T sound.
Е, е: Rendered the same as the Latin E, both having derived from the Greek letter epsilon (Ε, ε), but instead of making the familiar “eh” sound, it makes more of a “yeh” sound, as in “yet”.
Ё, ё: Although usually grouped with normal Cyrillic E, this makes a very different sound, “yo”. It isn't very common—and indeed, was introduced relatively recently in the history of Russian typography—and the use of it is by no means obligatory in media, which often merely uses the E to represent both sounds.
Ж, ж: Though this letter may appear frighteningly and conspicuously foreign, it merely makes a “zhe” sound, like in the English words “treasure” or “mirage”. This letter has no equivalent in either Greek or Latin, which makes its origins something of an enigma to linguists.
З, з: Not to be confused with the Latin number 3, this letter makes the familiar with Z sound except when before voiceless consonants, where it makes more of an S sound. This letter is derived from the Greek letter zeta, (Ζ, ζ).
И, и: Derived from the Greek letter eta (Η, η), this letter is almost perfectly equivalent in sound to the English I, as in “machine”.
Й, й: In Russian this is called the И краткое, literally “short I”, and is pronounced similarly to the consonant form of the English Y in words like “toy”.
К, к: Pronounced just like our Latin K, no tricks about it. This letter is derived from the Greek kappa, which looks precisely the same.
Л, л: This letter is almost a direct equivalent to the Latin L, making it pretty simple to pronounce: “el” as in “yellow”. It originally comes from the Greek letter lambda (Λ, λ).
М, м: Originally derived from the Greek letter mu (Μ, μ) with no difference in pronunciation between it and its Latin counterpart M.
Н, н: This one is something of a false cognate, making the same sound as our Latin N, however much it might resemble our H—even though both are derived from the Greek letter nu (Ν, ν).
О, о: This is one of the more difficult letters to pronounce in Russian, its sound depending on its position. After hard consonants it's comparable to the typical English O, like at the beginning of “orangutan”, but in unstressed positions it takes on akanye, or vowel reduction. The best way to explain this change is by the shift from an “oh” sound to an unstressed “eh” or “ah”.
П, п: Though formally referred to as “pokoy”, this letter is pronounced precisely like the Latin P. It originated from arguably the best known of the Greek letters, pi (Π, π). This is not to be confused with the previously discussed Cyrillic letter Л, which has a curved line in front as opposed to a straight one.
Р, р: This is another confusing letters for users of the Latin alphabet. Though it looks precisely like our P, it is pronounced like our letter R, “ehr”, albeit with a subtle, airy trill. This letter was developed from the Greek letter rho (Ρ, ρ).
С, с: As oddly as the last letter, though this exactly resembles our C, it is pronounced more like our S, “es”. The Latin and the Cyrillic letters are derived from the Greek gamma (Γ, γ) and sigma (Σ, σ), confusingly and respectively.
Т, т: This letter is a direct correspondence to the Latin T, pronounced “teh”. It developed from the Greek letter tau (Τ, τ).
У, у: More confusion yet: this letter, while appearing precisely like the Latin Y, is actually pronounced with an “ooh” sound, generally corresponding to the Latin U. Derived from the Greek letter upsilon (Υ, υ).
Ф, ф: It may look exotic, but this letter really just corresponds to our letter F, so pronounced “ef”. It is originally derived from the Greek letter phi (Φφ). However, this letter—and indeed, this sound—is found almost exclusively in words of foreign origin.
Х, х: While this letter appears rather like our Latin X, it makes a somewhat strange sound. The equivalent in English has largely died out, existing only in local pronunciations of words like “loch”. The best way to attempt this pronunciation is “kha” sound, with the “k” blending into the “a” as much as possible. It originated from the Greek letter chi (Χ, χ).
Ц, ц: Another letter with peculiar pronunciation: think of the pronunciation as being like the “ts” in “cats” and “sits”, a “tse” sound. This letter is thought to have derived from the Hebrew letter tsadi, צ.
Ч, ч: This letter makes the“che” or “cha” sound, as in the English “change”, except for a few very particular exceptions.
Ш, ш: This letter is roughly equivalent to the English “sha” sound, with no Latin alphabet equivalent—indeed, it is speculated that a separate Cyrillic alphabet was created precisely because it had no letter with which to make this sound.
Щ, щ: This letter closely resembles the previous in both appearance in pronunciation, sounding something like a cross between the English “sh” and “ch” sounds, or between the Russian Ш and Ч—think something like a “shta” or “shcha”. This letter is formed from combining the T and Ш letters, which makes sense given the sound it makes.
Ъ, ъ: This letter, “yer”, known as the твёрдый знак, or “hard sound”, because what it does is not make a sound in its own right, but “de-palatalizes” the preceding consonant, keeping it hard.
Ы, ы: “yery” or “yeru” as this letter is called, is virtually the same as the “i” sound of the previously discussed letter И, except that this one is instead placed after hard consonants, which results in a slightly shorter pronunciation.
Ь, ь: Known as the “soft sign”, this letter is the counterpart of the previously discussed Ъ, the “hard sign”. Just as with Ъ, this letter makes no sound in its own right, instead modifying the surrounding consonants by palatalizing them. This letter is also occasionally is used to indicate separate, distinct pronunciations of surrounding vowels.
Э э: Called Э оборотное, literally “backwards E”, this letter really is precisely the equivalent to the Latin E—indeed, Э originally developed as a variant on E. Think of the pronunciation of the E in “met”. Be careful not to confuse it with the similar З, or with the number 3.
Ю, ю: This letter is best equivalent to the Latin U sound. It is thought to have originated as a merger between the Greek letters omicron and iota (οι).
Я, я: This sound makes the palatalized version of the Latin A, to be compared with where we began: the—Cyrillic—letter А.
А, а * will take you to the Wikipedia page on the Cyrillic letter A. On the right hand side of the Wikipedia page are links to the other letters of the Cyrillic alpahabet, if you are interested in additional information of the history of the letters.