Yes, you always capitalize the first letter of the first word of any sentence. Yes, you also capitalize the letters in proper names: "Barack Obama", for example. This is a pretty good guide: http://www.grammarbook.com/punctuation/capital.asp

Capital letters are definitely important. Ignoring the rules makes your writing look careless and lazy. Remember that English has its own rules for capitalisation (eg. the names of languages start with a capital letter in English, whereas in a different language you might not use a capital), so do look at Tom's link and learn how it's done.

As Tom said, it is important to capitalize the first letter of any sentence. You should also capitalize people's name (for example: Sara), place names (for example: China or Beijing) and brand/business names (for example: Coca Cola). It is also common practice to capitalize the main words of a job title (for example: Bank Manager) but this is not necessary. I hope that helps :-)