If you look at the history of human civilizations encountering each other for the first time, generally, when one is significantly more advanced than the other, the end result is catastrophic for the civilization with the lower level of technology. Jared Diamond's Pulitzer-winning Guns, Germs and Steel is largely about this idea.
If you wanted to do a story where things went pretty smoothly for humans in an alien invasion scenario, what you'd need to rely on is the idea of the psychology of the very-advanced invaders being radically different from anything we see in analogous human conflicts. This is obviously pretty wide open territory to explore. You could make that difference about the aliens' culture, their biology, or just the fact that they've survived long enough to build spaceships capable of traveling so far and thus moved beyond the attitudes of humans.
On the other hand, if you wanted to directly model an invasion story of the history of cultural conquests on Earth (which, by the way, I'd find pretty cool) major factors you'd want to think about are disease, human superstition, opportunistic human behavior, and the tendency of low-tech cultures to profoundly underestimate high tech weapons/technology.
You'd also want to carefully think about what specifically the aliens want on Earth, and how they plan to go about getting it. If you're talking about a small group of alien scientists wanting to take core samples in Antarctica, wise cooperation from Earth governments might minimize the impact of the entire event. If you're talking about a situation that puts aliens in close, prolonged contact with the human population (making use of human labor or evacuating populated areas, for example) then you'd have human resentment and fear combined with poor understanding of alien weaponry, possibly resulting in violent skirmishes with devastatingly one-sided results. Less obviously, think about how complex our economy is and how reliant we are on having existing networks of transport and trade intact. Our alien invaders might not be interested in exterminating us outright, but there's a lot to carelessly mess up in our human ecosystem.
As for the long term, realistically, the Native American analogy you put forward seems like it could make for a plausible story, but do keep in mind that exposure to European diseases was a massive factor in that case; it's far easier for cultures to collapse when they've already had their numbers massively reduced by a sweeping, extremely contagious and deadly virus. Then again, deadly virulent disease being spread on first contact between cultures is the norm, not the exception on Earth, so we might encounter something similar with aliens, depending on their biological similarity to us (or perhaps they're just teeming with deadly infectious nano-bots).
Without an element of disease or warfare dramatically reducing the size of our populations, we'd probably hang onto more of our cultural identities for longer, but in the end, we'd all pretty much break down into predictable groups: those who tried to fight the aliens (dead), those who sought opportunity to use the chaos of the invasion to get ahead (rich), those who sided early and worshipfully with the invaders (depends, but rarely ends well in human history), and those who kept their heads down and quietly adapted (alive but maybe unrecognizable in a few centuries, especially if this is an alien species we're reproductively compatible with).
