In most Russian dialects, о is pronounced as in hello only when it is stressed. If it comes immediately before the stressed syllable, it is pronounced 'uh', as in money. Otherwise, it is pronounced 'ah', as in father. For example, the word "хорошо" is stressed on the third syllable, so the word is pronounced 'kha-ruh-sho', not 'kho-ro-sho'. Notice that the first "o" is pronounced as "ah", the second as "uh", and the third as "oh". However, if you pronounce о as 'oh', Russians will still understand you.
The suffix -ого is usually pronounced 'ovo', not 'ogo', and -его is pronounced 'yevo', not 'yego'.
Consonants can be either hard or soft. They are hard by default, unless followed by the vowels я, е, и, ё, or ю, or the soft-sign ь. Because of this, consonants rarely need to be made hard, so the hard-sign, ъ, is rarely used. Hard consonants are so called because they're short and brief, whereas soft consonants are just softer versions of their hard counterparts. For example, the hard consonant 't' in the English word tank is pronounced with the tip of tongue on the ridge of the hard palate - but if you pronounce it with the tip of the tongue on the back of the upper teeth, the 't' sound becomes soft.
At the end of a word, or before unvoiced consonants (like T), the following consonants change pronunciation: б→п, в→ф, г→к, д→т, ж→ш, and з→с. Notice that these are fairly similar, and since they make for easier pronunciation of the word, they are easy to predict. For example, ход is pronounced khot, not khod.
Palatalisation (or soft consonants) means the letter is pronounced with the middle of the tongue. Ex: Pronounce ball while holding your tongue.