Before you can start learning any vocabulary or grammar, you need to be able to pronounce the words you will see.

Vowels

A / Á / Â	Always long, as in 'father', never short like in 'hat'
E / É / Ê	This is always like the e in 'fed' or 'head', never 'feet'
I / Í / Î	If it starts a word it is a 'y' sound as in 'yes' (apart from im and i as these words are too short). An 'i' without an accent is pronounced short, as in 'sick', or 'hit'. An accented 'i' is a long 'ee' sound, like the 'i' in 'machine'
O / Ó / Ô	A short o is like the 'o' in 'dog' or 'lot'. An accented o should be pronounced like the O in 'snow', 'oh'
U / Ú / Û	Pronounce this like the 'oo' in 'tool'
Y / Ý / Ŷ	This should be pronounced like a rounder version of a 'u', as in the French 'lune'

Consonants

C	This is always hard as in 'kite', never soft like the English 'cent'
CH	Never as in 'church', always as in 'loch' or 'bach' from Scots/German
DH	The voiced 'th' sound as in 'these', or 'those' (not 'thing' or 'thief')
F	Beginning of word; soft as in 'father'. Anywhere else in the word; a 'v' as in 'verge'
G	Always hard as in 'get', never soft like in 'edge'
HW	The older British English pronunciation of 'white', or with a slight breath before the 'w' as if the letters WH in 'white' were reversed
I	As mentioned above, it becomes a 'y' as in 'yacht' if before a vowel or starting a word
L	Clear as in 'let' or 'leaf', never 'dark L' as in British English 'bottle'
LH	The Welsh LL sound (as in 'llan'). For an approximation, pronounce this as if the letters were reversed and written HL
NG	At the beginning or end of a word, as in the English 'ring' or 'sing', otherwise as in 'finger'
PH	An 'f' sound as in 'phonic' or 'phone'
R	Trilled, as in Spanish
RH	At the beginning of words only: This is what's known as a voiceless trilled R (for a rough guide pronounce it as if the H and trilled R were reversed, it should sound quite breathy with a slight trill)
TH	A soft 'th' as in 'thing' or 'thick', never as in 'the' or 'that'
W	At the end of a word you have two options for pronunciation. The Elves pronounced this like a W in 'woe', 'wight', but this is very difficult for a Man to replicate as you have to be exceptionally careful not to add in a 'schwa' (the vowel sound of the ER in 'butter', 'brother', or the AR in 'wizard'). The Gondorians pronounced a final W like a Sindarin U. Anywhere else in a word a W should be pronounced as in 'wet' or 'woe'

Diphthongs

AI	Like the 'i' in 'fine' or 'light', not like the word 'straight'
AE	Same as for AI, the 'i' in 'fine' or 'bright' - see below for further information
AU	Pronounce like the 'ou' in 'loud' or the 'ow' in 'now'
AW	As for AU above, pronounce like the 'ou' in 'loud' or the 'ow' in 'now'
EI	This is like the English 'ay' in 'play' or 'say'
OE	This is the 'oy' sound in 'boy'
UI	Pronounce this like 'ui' in 'ruin'

Vowel/Consonant combinations

ER	As in the word 'fair'
IR	As in 'dear' or 'ear'
UR	As in 'poor'

Remember that in all of these, the R is still trilled

Pronunciation of vowels with accents

As you should have noticed, the vowels have variants with accents: A / Á /  - these denote the length of the vowel, going from shortest to longest, left to right - an  should have approximately twice the length of an A.

Pronunciation of i

You may have noticed the way I have described the pronunciation of the vowel 'i' is a little different from the usual way people tell you to pronounce it. This is due to a set of linguistic notes in The Road Goes Ever On: A Song Cycle, by Donald Swann. Rather unhelpfully these notes aren't published anywhere else, but due to the date on which they were written they supersede what was published in the Appendices of the Lord of the Rings. These notes detail how the vowels should be pronounced, and read as follows:
"The short vowels may be rendered as in E. [English] sick, bed, hot, foot (for u), though o is intended to be rounder than in modern E."

A further note on AI and AE

These two sounds are probably two of the hardest to try and describe, although the above is your starting point!
AE and AI should not sound identical to each other - instead the sound should start off as the 'i' in 'light' or 'bright' and then glide onto the second vowel, producing a sound that should sound something like
AE-E = the 'i' in bright followed by a Sindarin 'e' in 'fed'
AE-I = the 'i' in bright followed by a Sindarin 'i' in sick or 'ee' in machine (if accented)

A further note on TH

When TH occurs in a compound (two or more words that have come together to form a new word) the two sounds are pronounced separately as a normal T followed by a H.
Examples:
Lanthir = Waterfall (Lant + sir)
Panthael = Fullwise (Pant + sael) - a nickname for Samwise Gamgee

Syllable stress

Tolkien's note on this can be found in Appendix E of The Lord of the Rings, but I shall summarise it here too:

Number of syllables in the word	Where the stress falls
Two	On the first syllable (EL-rond, AR-wen)
Three or more	On the penultimate (second to last) syllable (i-SIL-dur, peri-ANN-ath) or on the third to last syllable (gal-AD-ri-el)

Long vowels: A, U, Y, any accented vowel, vowel/consonant combination or diphthong
Short vowels: E, I, O

As you can see from the table above, longer words have two different patterns. How then do you know which syllable to stress in these longer words?
Penultimate syllable stress:
This is used when this syllable contains a long vowel. We also use this pattern if the syllable is a vowel (any vowel) followed by two or more consonants.
Attested examples:
Isildur (i-SIL-dur). Although the I in SIL is a short vowel, it is followed by the consonants LD, and so the stress falls on SIL.

Third to last syllable stress (antepenultimate):
This is used when the penultimate syllable contains a short vowel or any other vowel followed by just one consonant.
Attested examples:
Denethor (DE-ne-thor). NE is the penultimate syllable, but it contains the short vowel E, and so the stress falls back onto DE.
Boromir (BO-ro-mir). RO is the penultimate syllable, but it contains the short vowel O, and so again, the stress falls back onto BO instead.
Galadriel (ga-LAD-ri-el). RI is the penultimate syllable, but although it is followed by the vowel E, they are not a diphthong as they are two separate words so the I in RI remains short - thus the stress must switch to the syllable before it.

Possible exceptions to these rules:
In Tolkien's own reading of A Elbereth Githoniel (linked below) he distinctly stresses the word Linnathon as LINN-a-thon. This may be because the stress falls on the first part of a verb, or because he simply ignored/forgot his own pronunciation rules! As we cannot know for certain, I would suggest you follow the rules above.