Visit Sanskrit websites. You'll find a number of websites that show the Sanskrit alphabet along with the sounds' Roman equivalents. Copy or print these lists and try to memorize a few letters and their associated sounds at a time. Bear in mind that the Sanskrit alphabet is not analogous to the Latin one: it has more than 26 symbols and very different sound constructions. A complication with Sanskrit is that it can be written in more than one alphabet. The most common alphabet is the Devanagari alphabet. If this is not useful to you, you should also download the alphabets and Latin equivalents for the Grantha, Sharda amd Siddham alphabets and learn these in a similar way. All these alphabets and their transliterations are available at omniglot.com.
Transliterate a phrase. Once you feel you have a good idea of the Latin alphabet equivalents of the Sanskrit alphabet, try taking a simple phrase which you can access in both Devanagari and its Latin transliteration. Without looking at the transliteration, test your knowledge of the Sanskrit in the phrase and transliterate it yourself. Then compare your transliteration to the one you already have and see if you made any mistakes. Do this for a number of different phrases, correcting yourself each time.
Read Sanskrit passages aloud. Although you won't be able to test yourself -- unless you happen to know a Sanskrit reader who can help you -- this will gve you an idea of how easily you are coming to recognise Sanskrit letters.
Take an on-line tutorial. You'll find a number of free on-line tutorials that teach you the basics of Sanskrit. Try Achayra at acharya.iitm.ac.in for a set of 12 free lessons, or UKindia.com for step by step Sanskrit lessons.
Buy a book or lesson kit. You may prefer to purchase a book that can teach you Sanskrit, or to purchase an interactive kit. The American Sanskrit Institute offers both books such as the Sanskrit Atlas, and CD-based tuitional Sanskrit courses.
Take a course. Sometimes the easiest way to learn a language is via a qualified teacher. Many universities with centres of Asian studies offer courses in Sanskrit, as does the American Sanskrit Association, which gives both weekend and immersion courses.