Hungarian Consonants sound more or less as they do in English. 

There are some exceptions, in particular Hungarian consonant clusters, cs, gy, sz and zs. These function as separate letters and usually do not occur at the beginning of a word. They are pronounced as follows:

cs as the 'ch' in 'church' - e.g. csütörtök -Thursday ( Days of the Week)

gy as the 'j' in 'jury', but lighter. Listen to gyógyszertár - chemist (Places and Buidings 1)

sz as the 's' in 'salt' - e.g. szerda - Wednesday and szombat - Saturday (Days of the week) 

zs as the 's' in 'pleasure - e.g. rizs - rice (Words to do with food. General)

Other different Hungarian consonants:

c as the 'ts' in 'hats'
j as the 'y' in 'yes'
ly more of a 'y' than an 'l' sound. Listen to hüvelykujj - thumb in Parts of the Body
ny as in the Russian nyet
r trilled as in Scottish
s as 'sh' in 'shop'
ty as the 'tu' in 'tube' or 'tune' in British English
w only occurs in imported words, and is pronounced as 'v'

When consonants are doubled, sound both elements, so 'butter' would be pronounced 'but-ter'. When consonant clusters are doubled, only the first letter is affected, e.g. a double gy becomes ggy.

Hungarian Vowels are less straightforward. In general vowels can be related to their English equivalents by remembering that an accent lengthens them. It is important to remember that an accent not only alters the sound, it can also alter the meaning.

A good section to visit is Countries and Where I'm From, in which 9 of the 14 vowels listed below can be heard. Directions 2 conveniently provides the full range of unaccented vowels.

a tends towards the 'u'in 'hut', 'but'
á sounds like 'a' in 'father'. Compare the two 'a's' in kanál - 'spoon' (Eating)
e short English 'e' as in 'get', 'bet', etc