Begin with the Arapaho alphabet. Arapaho has eight vowels: e, ee, i, ii, o, oo, u and uu. Additionally, there are three diphthongs (two letters that make a sound different than the sum of each): ei, oe and ou. Arapaho has only 12 consonants: b, c, h, k, n, s, t, w, x, y, ' and 3. According to native-languages.org, a double vowel (ee, ii, oo, uu) is not pronounced doubly, but is rather held longer. C is pronounced like "ch" in chair, x is like the "ch" in the Scottish "loch," ' is a glottal stop, and 3 is pronounced like "th" in thin.
Learn the tonal markings used in Arapaho. An acute accent over a vowel means it has a high tone: é, éé, í, íí, ó, óó, ú, úú. A caret (^) indicates a falling tone: ê; êê, î, îî, ô, ôô, û, ûû. Practice what you've learned about Arapaho's alphabet and tones using the University of Colorado's guide to Arapaho greetings (see Resources below). You can also get help with pronunciation from Berkeley's Audio Dictionary of Arapaho (see Resources below).
Use of the materials on the Arapaho language provided by native-languages.org and the University of Colorado. Both sites offer lessons in Arapaho as well as sample texts to read and check your skills. See References below for links.
Use a textbook to help you learn. Andrew Cowell and Alonzo Moss Sr. have written one of the very few books available on the Arapaho language, and it can be used as both a grammar reference and a language textbook (see Resources below).
Aim for about an hour's worth of study every day with a variety of materials. Especially when you are starting out, this may help you make quick progress.