How did the people of the Marquesas Islands (or others) make their way to the Hawaiian Islands? What techniques/special skills would they have used, what items did they take with them, how did they alter the environment they found there.

I can't answer all of this, but if you're interested in Hawaiian settlement I would highly recommend that you visit the Kahanu Garden on Maui. I've been to one of its sister gardens on Kauai, and they did a great job highlighting the native and canoe plants of the islands.

What are canoe plants, you ask? Well, this is where I answer part of your question; namely, what they took with them. To a certain extent, Hawaii was similar to the other islands settled by Polynesian voyagers. However, they didn't just "live off the land," as some romanticized depictions imply. They couldn't, because most of the islands they encountered didn't natively have a ready supply of edible and useful plants and animals. Instead, they brought plants and animals with them.

One of the most important canoe plants was taro, a root vegetable. It is cooked, then pounded into a paste to create poi, a staple of the Hawaiian diet. Water is often added to vary consistencies, and poi is also sometimes left to ferment a little, which gives it a vaguely yoghurt-like flavor.

Coconut trees were another very useful canoe plant. In addition to the obvious food usage, they could be used for lumber, and their fronds used for thatch. In fact, there's actually very little you can't make out of some part of the coconut tree, making it an important plant to have if you were venturing off to an uninhabited island.

Versatility was key in a canoe plant. Boat space was tight, and the plants and seeds were tenderly looked after. Unlike animals, which could be slaughtered if times were rough, saplings and cuttings didn't provide a great source of emergency food. So, many of the canoe plants were, like coconuts, used for multiple purposes. The leaves of the banana plant were used for roofs. The mastwood tree was great for building ships, but also produced an oil that could be used as medicine. Gourds were grown to be used for containers for just about everything, as well as drums and rattles for making music.

Anyone who has been to Hawaii knows that pork and pig products are a staple of their cuisine. Not only did the Polynesian settlers bring various plants with them, they brought pigs, chickens (junglefowl), and a type of dog that was once common to Polynesia. As you might imagine, they ate the pigs and chickens, as well as the chicken eggs.

As you might not imagine, they also ate the dogs. The so-called "Hawaiian dog" more or less roamed free in the villages and taro fields, and were primarily fed poi. They do not seem to have been suitable for hunting or any of the usual things dogs are bred for. Instead, they were primarily a food source, and also were seen as a sort of "good-luck" charm. The Hawaiian dog was one variant of the Polynesian dog that was once common across the Marquesas, New Zealand, and other Polynesian areas. They have all died off since, so we are unfortunately left primarily with depictions and bones. Cook described them as "having short crooked legs, long backs, and prickly ears.”

While feral pigs were (and very much still are) around in Hawaii, Polynesians seem to have generally tried to keep their pigs contained and close. We know this in part because free-roaming pigs would have been a major nuisance to the sweet potato and taro plants that were integral to the Polynesians' lifestyle. We also know this because Hawaii at the time did not have the resources necessary to sustain large populations of feral pigs. Even if they were a bit more free-ranging, the pigs would have had to keep close to humans if they wanted reliable food.

So, the early Polynesian settlers of Hawaii were quite prepared, and had a fairly sophisticated agricultural system that helped ensure their survival in their new home. I have used the past tense in this article, because you were asking about the first settlers, but it's important to recognize that Hawaiian and Polynesian culture is not a thing of the past. That culture is still very much alive today, and you will undoubtedly meet plenty of proud Hawaiians who may still know and use these same plants in the ways their ancestors did. You will certainly eat plenty of coconut and pig products! I'd recommend trying poi as well, not because I'm particularly fond of it (I'm not), but because it has been so important to Hawaiian culture. You'll find it in a variety of consistencies and tastes, some quite sour, so even if you don't like one version, you may like another.