Sanskrit is written in the Devanāgarī (देवनागरी) script, shared with Nepali, Marathi and a number of other Indian languages. Learning Devanagari is not quite as difficult as you might think at first glance, but mastering it takes a while and is beyond the scope of most travellers. See Learning Devanagari for a primer.
Most English speakers find Sanskrit pronunciation rather challenging, as there are 11 separate vowels and 35 separate consonants, employing a large number of distinctions not found in English. Don't let this intimidate you: for most of its speakers, Sanskrit is not a mother tongue, and many native speakers are quite used to regional accents and mangling in various degrees.
The key distinction is the difference between short and long vowels. In this phrase book, long vowels are noted with a macron (ā), which short vowels are listed without one. You will often come across non-standard romanizations, noted in parentheses below when applicable.
Many Sanskrit consonants come in three different forms: aspirated, unaspirated and retroflex.
Aspiration means "with a puff of air", and is the difference between the sound of the letter "p" in English pin (aspirated) and spit (unaspirated). In this phrasebook, aspirated sounds are spelled with an h (so English "pin" would be phin) and unaspirated sounds without it (so "spit" is still spit). Hindi aspiration is quite forceful and it's OK to emphasize the puff: bharti.
Sanskrit retroflex consonants, on the other hand, are not really found in English. They should be pronounced with the tongue tip curled back. Practice with a native speaker, or just pronounce as usual — you'll usually still get the message across.