Here is an answer from a British English perspective. Note that the American English usage isn't quite the same. 'Shall' is still used in formal situations in written language, for example in legal documents, and more formal business correspondence.
When it comes to spoken language, both 'will' and 'shall' are usually contracted to 'll, so you needn't worry about which word you are using. Rather than 'I will' or 'I shall', we normally simply say 'I'll'. The emphatic form, without the contraction, is 'I will'. The one situation where 'shall' is still common in spoken English, at least in the UK, is in question forms. Obviously, if you are using a question form, the modal verb has to come before the subject, so it cannot be contracted. We use the form 'Shall I....?' to make offers and suggestions:
'You look cold. Shall I close the window?'
'You'll be home light tonight, won't you? Shall I make dinner?'
This is similar to 'Should I..?', but there isn't the implication of obligation which comes with 'should'. It is simply an offer.
We also use 'Shall we...?' to make suggestions about a joint activity.
'You look tired. Shall we go home?'
'The weather's nice today. Shall we to the beach?'
Don't believe anybody who tells you that these forms are formal, pretentious or old-fashioned sounding. Anyone who says this has either missed the point or doesn't listen properly. All of these are very common, natural forms in British English. You'll learn them at elementary/low intermediate level within the function of 'making suggestions and offers in everyday language. '
Also, note that the sound is often slurred in relaxed speech. Just as 'Do you ....?' sounds like 'D'you' when native speakers are speaking in a natural way, the words 'Shall we...?' sound like 'Sh'we...?' There is nothing formal or archaic about saying 'Sh'we go now?'. It's just what we say. Watch any Harry Potter film, for example, and you're sure to hear someone say 'Sh'we...?' for a suggestion.