Always make your consonants at the end of the words UNVOICED. The ending consonants are all the time unvoiced in Russian. For example, a Russian would say 'as I know' like 'ass I know' which immediately distinguishes Russians (weird example, but it's from my experience and i remember it well).

Russians don't have clear distinction between your short and long vowels. The 1st vowel in 'cup' and 'carpet' would be pronounced similar. And this would be neither long nor short (something like you pronounce 'er' in 'teacher'). The same applies to all vowels, they are all neither long nor short, and also neither closed nor open.

To exemplify, i would suggest:

pen - pan
see - sick
sock - sort

No distinction is made by Russians here.

Concerning consonants, and meaning again they are unvoiced at the end, no distinction will be made between 'bed' and 'bet', they both will be pronounced in the same way, i.e. with 't' at the end and again the vowel will be like you'd pronounce 'er' in 'teacher'.

The most difficult sound for a Russian is 'th'. We don't have it. Make it either 'z' or 's', or even 'f'.

Nowadays, also Russians in words like 'go', 'no' tend to say [ou], not [eu].

It's a complicated topic you suggested, and it takes a lot more aspects, such as make your 'sh' as in 'she' or 'j' as in 'john' very hard as our equivalent sounds are not so soft as yours.

Hope, it will help you somehow, but to fully grasp it you will have to make a system of it, cos here there are some systematic differences and they really can be traced if you learn to hear it.