Konkani is one of the many languages used in India. It is spoken mainly in the states of Maharastra, Karnataka, Goa and Kerala. Though not as widespread as Bengali, Hindu or Urdu, Konkani is the official language of Goa. Konkani has been written in a number of scripts, with devanagari being the official script in Goa. Konkani is also used in other parts of India, resulting in several forms of the language. If you are interested in learning Konkani online, your choices will generally be between Goan Konkani and Mangalorean Konkani, with Goan being more prevalent. (References 1)
Mangalore is a city in the state of Karnataka. Konkani is a main language of Mangalore along with Tulu, Kannada and English (References 2). A quick guide to useful phrases in Konkani is the "Common Kannada, Tulu and Konkani Phrases" page at Mangalore.com. Short conversational phrases and sentences such as "How are you?" and "What time is it?" are written in English and then compared in all three languages. There is also a table of everyday vocabulary such as "doctor", "read", "why" and "money" on the same page. Mangalorean.com has an even more developed resource page. The Konkani Zone has six different lessons in Mangalorean Konkani (References 3).
Both sites transliterate Konkani by using the English alphabet, not devanagari, so knowledge of the script is not required, nor taught.
If you are more interested in the form of Konkani spoken in Goa, the Goa Holiday Homes website has a page full of everyday phrases and conversational vocabulary. Sections are divided into "Everyday Conversation," "Eating and Drinking," "Accommodation and Getting Around," "Shopping," "Relationships," "Time," "Days," "Months" and "Numbers" (Resources 2).
"Teach Yourself Some Konkani" at Goa World not only has a table of helpful phrases but a clickable interface that allows you to hear the sounds of the alphabet. The page also has three lessons, in which you learn vocabulary, see its usage in a sentence, and get translations in English of both the word and the entire sentence. If you master those phrases, the Our Karnataka website has even more. Our Karnataka's Konkani page has 36 lessons available.
If learning Konkani inspires you to tackle devanagari script, there is an excellent online devanagari script tutor at Hindibhasha.com. The script tutor covers writing consonants, vowels and conjuncts, and even provides quizzes so you test your understanding of both the script and the pronunciation.