So I have a group of characters who go to a new land. They're stuck on this land, and eventually encounter the culture. How can I show their gradual assimilation/adaption of traits/characteristics of the new country to the point where they don't speak their native language and their native customs are now foreign-seeming to them?
I'd say you just describe the details of the transition. It might help to look at how "going native" is handled by other authors. For example, James Clavell's Shogun (sailor becomes samurai), or Heinlein's Stranger in a Strange Land (human raised by aliens adjusts to Earth), or Kipling's Jungle Book (boy raised by wolves and animals (eventually) encounters his own people). There are many more.
This happens all the time in the real world. Immigrant populations are constantly having to assimilate with new groups of people. Most families have stories about odd things their grandparents or great grandparents did or words they used that were from the old country. It's likely that you won't have to work too hard to find someone close to you that is going through this experience right now.
When my great-grandfather died, my great-grandmother had my great uncle go out into the yard and bring in a big pile of dirt from the garden and put it on the living room floor. She then laid my great-grandfather's body on the pile of dirt. Nobody in the family had any idea why. They thought she was crazy. Turns out it was a cultural tradition from parts of Eastern Europe.
You might try reading through some of the posts on www.humansofnewyork.com. Sometimes you don't have to be from somewhere else to have a hard time fitting in.
Cultural assimilation usually goes both ways. There are recent archaeological findings where archaeologists are tying the sun worship cult in ancient Denmark to sun worship in Egypt. When people move from one place to another they pick up new things but they also bring things with them, but it usually takes a lot of time. Assimilation happens, but it's usually the children of the immigrants that assimilate and not so much the initial immigrants.
If you were going to be realistic, what you'd probably have is that your group of people would stick together while moving into the new culture. They might learn some of the new language and pick up some of the new ways of doing things, but it would be their children that would be the bridge. The children would be the ones that would kind of fit into both cultures but not really in either of them.
Having said all that about groups, there are the individuals that you'll see that move from their own culture where maybe they didn't ever really fit in and adopt a new culture almost as a way of creating a new identity. The movie Lawrence of Arabia is a good example of that, also C. J. Cherryh's "The Faded Sun" series. And also this recent Escape Pod episode, "Them Ships."
Culture and customs are not some uniform thing. Every family is a micro culture of its own with rituals, beliefs, and even family language that isn't shared outside the family. Toddlers invent words that their parents pick up and use in regular family conversations. I've found that the more children a family has, the richer their individual family language is. No one family, no matter how similar they are to their neighbors, celebrates Christmas or Hanukkah or whatever the same exact way, but they are consistent within their individual families.
So I hope I've answered your question and given you some ideas on how to proceed with this. Short answer - I don't think the original group of people would forget their culture and customs, but if they were separated from their native people for long enough, I'm betting things would have changed enough back in the old country to make the old country seem strange.
I'd pick a few things and highlight them.
Like, at the beginning of the story the characters arrive in this culture and comment to each other about some strange custom. Like, "Wow, these strange people wear socks with sandals. Why would anyone do that?" Then later you mention one of the characters putting on socks and sandals.
If you want to be subtle about it, you just say he did it and leave it at that, and leave it to the reader to realize, "Hey, just a few chapters back didn't this same character say that this was a bizarre thing to do?" If you're worried the reader might miss it, you can have another character ask him why he's doing this, and he replies, "Oh, I don't know, I didn't think about it. It just seemed normal." If you really want to make a point of it, you can have the character say, "Yes, now that we've been living here, I understand why the natives do this. You see ..."
If the point of the story is their cultural assimilation, then you could go through this for many customs and practices. If that's just a side note or a step to something else, then just mentioning two or three such things would probably be plenty. I'd intersperse them with other action. Like have the first one be something very minor, then other stuff happens, etc, and then the last one is something big, the Greek sacrifices his son to Moloch or something.