The dispersal of the Polynesians is archaeologically fairly recent. One suggested starting point is Solomon Islands or northern New Guinea. Carbon dating shows the pattern of setlement of the islands - I have no details of dates or order to hand, but it was into the central Pacific first, and outlying lands like Aotearoa, Hawaii, and Tahiti later.
All Polynesian languages are very similar (note Fijian is linguistically Melanesian, not Polynesian). By applying the consonant shifts you can almost read them off as each other, it seems: certainly they have many many common words and grammatical constructions in common.
Some (like Maori, Hawaiian) have W and others (like Samoan, Tahitian) have V: wahine ~ vahine 'woman'. The Hawaiian W is actually more V-like before some vowels.
Some have H, others have S. The main island of Samoa is Savai'i, which is obviously the same name as Hawai'i and Hawaiki. Presumably S is the older form but I won't swear to it.
The original K of proto-Polynesian changed into the 'okina. Maori preserves the original three consonants K T P.
However this is complicated by the fact that in some languages T has changed to K. So Hawaiian now has K and ' but no T. This is true in normal Samoan speech too, though T is still used in writing. Samoa's head of state Malietoa Tanumafili is pronounced Maliekoa Kanumafili except in formal settings... according to my Teach Yourself Samoan. This indicates that the change happpened before Hawaiian was set down in writing, but after Samoan was.
The Polynesian family is part of a larger group called Austronesian (formerly called Malayo-Polynesian). This includes all the languages of Indonesia, the Philippines, Micronesia, and Melanesia (but not New Guinea) as well; and also Malagasy of Madagascar; and also the aboriginal (non-Chinese) languages of Taiwan.
Archaeological evidence is matched by linguistic evidence (common words for kinds of canoe, animal etc.) for the original homeland of Austronesian speakers as being in Taiwan. From there their spread down into the Philippines, into Indonesia, and across the Pacific (and in the case of Malagasy across the Indian Ocean) can be dated with somewhat more accuracy than old language movements usually can.
