Spanish colonization of Hawaii, assuming here that someone stumbles upon it in the mid-sixteenth century, would probably mirror that of the other islands it controlled in the Pacific, like Guam. Hawaii would be a place for ships in the Manila-Apapulco galleon routes to stop and restock on supplies. Outside of that, the islands won't really be very important for the Spanish Empire and would otherwise be ignored. Not many Castilians came to the more attractive areas of the Spanish East Indies like the Philippines IOTL so I do not expect many to consider Hawaii to be a destination for Castilians. As for Filipino and Mexico immigration, you can expect to see a lot of it since it would serve as a midway point between the Philippines and Mexico, eventually bringing a synthesis of language, culture and cuisine onto the islands. Expect to see the native Hawaiian language to carry a lot of Tagalog, Spanish and Nahuatl loan-words from the migrants. You might reach a point where a significant minority of the native Hawaiian language comes from these three sources, not to mention whatever from the languages spoken from other migrants to the islands, maybe some that came in OTL like the Japanese and Chinese.
The native Hawaiian population will suffer a lot from diseases introduced by the Spaniards but since Hawaii is rather distant, I do not expect them to suffer the same fate of the native Americans and be completely shut out. Culture and language would be severely repressed by the Roman Catholic Church and the Spanish authorities. Resettlement might be practiced as a policy to make sure that the natives are all under the watchful eye of the Spanish garrison. What will remain of the upper class in Hawaii will become integrated and intermarry with what few Spaniards that arrive, forming the criollo population. Native Hawaiians will intermarry with the Mexicans and Filipinos that arrive on the island as soldiers, fishermen, whalers, merchants, farmers and workers in the large plantations, producing a variety of things like sugar, maize, rice, bananas, etc.
Catholicism would be the dominant faith of Hawaii though some indigenous traditions will survive, either by remnants practicing it in isolated rural communities or by missionaries who aren't so strict and allow them to continue, albeit in a rather Christianized setting, much like elsewhere. You'll likely have some Hawaiian Catholics preferring to God the Father by Wakea, etc.
Independence and how they achieve it is probably unknown and you can pretty much go anywhere with your imagination as how they might do it. It is likely that any independence movement will be headed by the criollos or at least have some criollo involvement, if we're murdering butterflies, sometime around the early nineteenth century like the other Latin American countries. Something like the Plan of Iguala (the part of having a European monarch serve as the constitutional monarch of Hawaii) might be attractive to the Hispano-Hawaiians to legitimize their desires for independence and perhaps even put them under the protection of a greater power, even leasing one of their ports.