Only two things in the world are more difficult than [French] pronunciation: [French grammar] and, of course, [English pronunciation].

First I'll give you a few hints to hide your [Texas|Texan] origin. Then you'll find, for your convenience, an alphabetical list of letters and how to pronounce them in French.

Please, do not pronounce "é" as "ay"

A very easy way, for a Frenchman, to make fun of [Briton]s or [American]s is to add the sound "[y]" everywhere. It's such a common mistake that I suppose teachers spread it on both sides of the Atlantic Ocean.

"[é]" is what the [International Phonetic Alphabet] writes [e], i.e almost like the "e" in "very" but with the mouth less open.

This mistake illustrates a fundamental principle, which in my opinion should help you to understand how French pronunciation works:

French vowels are simple, while English vowels are composite.

In particular, the French language has no [diphthong]s. Simple here doesn't mean easy. It means that only one flat sound, one [phoneme], is involved in the utterance of a [vowel]. Therefore, "é" does not end with some glide to "y". It may last as long as you want, but the sound at the end of the vowel is the same as it was at the beginning.

In English, on the contrary, one written vowel often corresponds to two phonemes. The obvious case is the diphthongs: "sound" ([a]->[u]), "twice" ([a]->[y]). This is also true of many other vowels, which sound gradually glides to another sound (typically "y"). See other examples in [IPA/ASCII Representation of Sounds in English]. Thanks [Mercuryblues] for his indications about this issue.

In the same way, a French [sentence] is very flat. In English, you need to put a [stress] on some syllable and half-pronounce others. In French, you pronounce most syllables (apart from the case of the [mute] e, see below); they all have approximately the same weight. In normal speech, you should stress only the last [syllable] in a group of words, even in a very long word like [anticonstitutionnellement|anticonstitutionnelle'ment]. Example:

'French 'vow[e]ls are 'simpl[e], while 'English 'vow[e]ls are 'comp[o]sit[e]
Les voyell[e]s fran'çais[e]s sont simpl[e]s, tandis que les voyell[e]s an'glais[e]s sont compo'sées.

However, ancient French did contain diphthongs. That's the reason why the written language contains so many [digram]s, ie. groups of vowels which correspond to only one phoneme.

Miscellaneous

Double consonants are almost always pronounced like one [consonant]. "belle" and "bêle" sound exactly the same.

A consonant other than c, r, f, l (tip: careful) at the end of a word is usually mute. We will see exceptions below. Example:

[clef]: klé (key, also spelled "clé")>

Non-French words. Some languages modify the [spelling] of foreign words to match the foreign pronunciation (fútbol in [Spanish]). Others keep the foreign orthograph but pronounce it as if it was a local word ([English]). Modern French usually keeps both the foreign orthograph and, as much as possible, the pronunciation. More and more French people pronounce the [jota] in Spanish names although this sound normally doesn't exist in French.
However, the French knowledge of foreign languages has well-known limits. Furthermore, stressing will always be done the French way, i.e on the last syllable. So "week-end" ([weekend]) will be pronounced "wik'ènd" (while a pure French pronunciation may be "wékan" or "vékan", and a pure English pronunciation would be something like "'wiikend").

Written accents. We will see below the accents that indicate the pronunciation ([é], [è], [ê]). Other accents ([à], [ô], [û], etc) only serve to avoid ambiguities in written language (a vs à, eu vs eû...). Or they indicate the etymology of the word. In either case they have no effect on pronunciation.

Alphabetical list

Now, we may examine in detail how to pronounce written French words. Here is an alphabetical list of letters or group of letters that sound differently in French and in English.

a, â, à: usually pronounced /a/ as in father. Examples: [mari], [pâtre]. Your teacher may tell you that it can be short or long, open (pâtre) or closed (mari). Nowadays, there is no real difference. Don't bother about the vowel length in French, unless you want to sound like a Parisian radio speaker from the 1950s.
ai: pronounced /e/ or /ε/ (see è below). I have already seen rules and lists of words about the pronunciation of "ai" in verbs, at the end of a word, etc, and I usually disagree with them, probably because I come from southern France. So don't bother too much studying whether a particular word (such as "j'ai", "aimer", etc) should be pronounced /e/ or /ε/. As long as you don't add a "y" at the end, it will be fine.
Examples:
j'ai: jé (I have)
j'aime: jèm (I love), but [aimer]: émé (to love).
Exception:
vous faisez: vu feuzé (you do)
aim, ain, am, an, em, en, im, in, om, on, un ([nasal vowels]): after a group of vowels, the letter m or n is not pronounced but adds [nasalization]. So "ain" equals to "ai" (see above) + nasalization, which is the same as "in". Exception: "un" equals to "œ" (see "eu" below) + nasalization.
There are four nasal vowels in French: an, in, on, un. Try to feel the differences in the following phrase: "un bon vin blanc" (a good white wine). [Parisian]s tend to confuse "un" and "in", but they have always sounded very different to my [Aquitaine|southern] ear.
Nasalization only occurs if the m or n belongs to the same syllable as the vowel, which is usually true when it is not followed by another vowel. Examples:
ami: a-mi (no nasalization)
emporte: an-porte (nasalization)

A double "m" or "n" following a "e" will be divided into two syllables, the first with nasalisation; after another vowel, it will be considered as a single consonant in the second syllable. Examples:

emmener: en-mené (to bring)
ennuyeux: en-nuyeu (boring)
année: a-né (year)
innocent: i-no-cent
au: see o below.
c: the rules are the same as in English, i.e it's pronounced /s/ before the vowels /e/, /i/ and /y/ , and /k/ otherwise.
Exceptions:

second: seugon (2nd) (as a side note: in theory, both "second" and "deuxième" mean "2nd", but "second" is normally used when there are only two elements in the list: the "seconde guerre mondiale" may become "deuxième" if a 3rd World War occurs one day...).
seconde: seugond (1/60th of a minute)
ç: the cedilla is used to force the letter to sound /s/ even before /a/ , /o/ or /u/.
cc: /ks/, if followed by a vowel. Examples
accepter: aksepté
accroître: akrwatr
ch: pronounced almost like /sh/. French speakers, when they hear the English word "[cheers]", hear "tchirz". Try not to pronounce the "t", and you've got it...
When followed by a consonant (r, l), and also in a few other words, ch is pronounced "k". Examples: choeur, chrysanthème, Chloé...
e: the most difficult vowel. It has a lot of different pronunciations, but most of them can be derived from rules based on the context:
- an ending "e" (or "es" in the plural form): it is usually mute (unless you are in southern France):
  sagesse: sagès (wisdom)
  If the word only contains one syllable, the "e" will be pronounced like "eu" (see below): le (the), ne (not), ce (this), que (that, which) etc...
  However, even in that case, the vowel may drop in the middle of a sentence. Example: celui que j'ai: seluikjé (or even "suikjé").
- e, followed by "mm": may be pronounced like "a" (femme: fam; fréquemment: frékaman), or like "am" with nasalization (emmener), or like è (lemme: lèm). The 3rd rule seems more natural to me when I try to imagine how non-existing words (cemme, remmonter...) could be pronounced.
- e, followed by another double consonant: usually sounds like "è". Example: belle: bèl (beautiful, feminine)
- e, followed by one consonant: may sound like "é" or "è". Usually sounds like "è" when followed by a non-mute consonant at the end of a word (see below). Examples:
  et: é (and)
  perdu: pèrdu (lost)
  appel: apèl (call)
- e, followed by d, r, t, z at the end of a word: "e" will sound like "é", and the consonant may be mute. Examples:
  pied: pié (foot)
  any verb ending in -er: manger, crier, lever...
  et: é (and)
  nez: né (nose)
- é (e acute): closed e (see the section on how not to pronounce "é" at the beginning of this writeup). IPA: /e/.
- è (e grave): sounds like "e" in "pet" (open e). IPA /ε/.
- ê: works like "ai" above. Say either "é" or "è". Examples:
  prêt: pré (ready)
  même: mèm (even)
- eau: see o.
- ei: "é" or "è".
  Exception: in cueillir, the "ill" rule prevails: keuyir.
- en: see an above.
- eu: /ø/ (closed) at the end of a syllable, /œ/ (open) when followed with a consonant. /ø/ is approximately the vowel sound in "first". Examples:
  peureux: pørø (afraid) (the first syllable ends before the "r"
  peur: pœr
  deux: dø (two)
  neuf: nœf (nine)
  Exception:
  
  eu: pronounced like "u" (past participle of "avoir")
g: same as j before e, i and y. Same as the g in "green" before other letters.
Exception:

vingt: vin (with nasalisation) (twenty)
gn: /ny/ (as in canyon).
h: mute, but may have an influence on links with other words.
i: /i/, as in "pit". /j/ (semi-vowel) before another vowel. Examples:
pitié: pitjé (pity)
calendrier: calandrijé (calendar)
ï : /y/. Example: aïe: ay (ouch)
ill + vowel, or vowel + il + end of word: /y/. Examples: ail (ay), maille (may).
in: see an above.
j: when the French hear the English word "jam", they hear "djam". Do not pronounce the "d".
o: may be open or closed. However, here again, it depends as much upon your origin than upon the word.
Exception: monsieur (mister) is pronounced "meussieu".
œ: see eu above. Examples:
œdème: ødèm
œil: œy
oi: /wa/, like in the English "want". It's not technically a diphtong, since "w" is considered a semi-vowel. Examples:
poids: pwa (weight)
on: see an above.
ou: like the English "u" in /boot/ but maybe a little shorter.
p: like in English. Exception:

sept: sèt (seven)
qu: always sounds like "k".
r: read [How to pronounce a French "r"].
s: "z" between two vowels, "s" otherwise, and usually mute at the end of a word. Examples:
dessert: déssèr (desert, the one that you like hot)
désert: dézèr (desert, the one that you don't like hot)
procès: procé (trial)
sc: sounds like s.
sh: sounds like "ch". Very rare.
ss: always pronounced "s" (never "z").
th: pronounced like "t", of course.
ti + vowel: usually "si". Examples: action: aksion (action)
Many exceptions:

entier: antié (entire)
tiers: tièr (third), etc.
u: /y/, like "ü" in German (über).
un: see an above.
w: either "w" like in English, or "v", depending on the word. This letter is rarely used in French. Examples:
wagon: vagon
week-end: wikend
x: /gz/ between e and a vowel, otherwise /ks/ (same as in English, I think). The "x" in "xc" is pronounced "k". Usually mute after u at the end of a word. Examples:
axe: aks (axis)
excellent: èksélan (excellent)
exécuter: ègzécuté (execute)
extraterrestre: èkstratérèstr (alien)
oiseaux: wazo (birds)

Exception:

six: sis (six)
dix: dis (ten)
y: /y/ before a vowel, otherwise /i/. Examples:
yeux: yeu (eyes)
payer: payé (to pay)
Cythère: sitèr
z: like z in English. Often mute at the end of a word. Examples:
nez: né (nose)
riz: ri (rice)

Of course, since we are speaking about the French language, there are many other exceptions, for example in [popular pronunciation]s: je ne sais pas: chépa (I don'k know, or I dunno) ouais for oui (yes)

Another approach of French pronunciation (more academic and a little less real in the way it differentiates open and closed vowels): http://www.frenchlesson.org/grammar/pronun/. I have used some of their examples.

Many thanks [Mercuryblues], [sloebertje] and [tdent] for corrections!

French pronunciation (thing)