That's how they used to teach native kids in Russian schools some years ago.
All vowels can be hard and soft except few. (Notice the use of "hard" and "soft" instead of "velarised" - "palatlised". I am not sure if the terms were avoided to make the rules more intuitive or because the kids were native and they didn't need more generic linguistic terms.)
Consonants are read "hard" or "soft" depending on the following vowels or special letters: hard and soft signs. The default state is "hard", so when a consonant is not followed by anything it is "hard", thus usage of hard sign to make consonant hard is obsolete in modern written language, but can be found in books dating before 1918.
Vowels that make consonants soft: e, ё, и, ю, я. Soft sign: ь.
Vowels that make consonants hard (or rather don't make them soft): э, о, ы, у, а.
- м in мера - мэр
- т in тёр - топ
- н in ныть - нить
- л in люк - луг
- р in ряд - рад
- с in нос - ось
The following consonants are always hard: ж, ц, ш.
The following consonants are always soft: й, ч, щ.
(De facto these might be not completely true for certain dialects or variants of pronunciation: e.g. вожжи, дрожжи, дождь might be pronounced with "softish" ж).
Hard and soft signs also have "separating" function. In this case the soft sign may loose the softening function: льёт - soft л, but вьюн - hard в.