Basically, и comes after a soft consonants (which are indicated by either a soft sign ь or the vowels я, ё, ю, и and е) and ы comes after hard consonants (which are indicated by the perceding vowel, i.e. а, о, у, ы and э).
Never use ы after the velar consonants г, к, х or the hushes ж, ш, ч, щ; use the vowel и instead. This is called the 7-letter rule.
If you don’t know what palatalization is, it means that the consonant is produced with the body of your tongue pressed against the hard pallate (http://goo.gl/grwypl). This is very important in Russian since the language uses it to differentiate between words.
Try to here the difference between:
·угол ‘corner’ http://goo.gl/5CBXCq and уголь ‘coal’ http://goo.gl/2yJauB. ·(он/оно/она) ест ‘(he/it/she) eats’ ·the б in быть ‘to be’ http://goo.gl/rlDvGQ and бить ‘to beat’ http://goo.gl/vk4D87. ·шест ‘pole’ http://goo.gl/YGqViM and шесть ‘six. http://goo.gl/7MIfyo.
Just ask if you have any questions!
After Г, К, Х, Ж, Ч, Ш, Щ, you always use И