In the very near future, humans (somehow) travel to a distant planet - and discover aliens who turn out to be uncannily like us. Their appearance is, to put it bluntly, freakish, but they are humanoid bipeds, and their life cycles, family structures, and basic needs are all reminiscent of human biology, albeit with the occasional surprising quirks. Their civilizations appear - at least at first glance - to have technologies, religions and political institutions analogous to bronze and late stone age cultures from the ancient Near East. It is not immediately obvious at first contact whether the differences run deep or are best said to be merely contingent.

Naturally, the very first thing the humans conclude is that these strange creatures are a veritable goldmine for the social sciences, and so they drop in teams of socio-cultural anthropologists (and perhaps also a scattering of psychologists, sociologists, economists, etc,) to do field research. The aliens prove very accommodating to all this... at least for now.

Given current academic practices, how do these anthropologists first go about it? Most especially, what are the major questions to ask, and models to test? What is the likely order of priorities, and what (aside from the blunt fact of meeting aliens, anxiety over whether contact is even desirable, and excitement no longer having only one species to study) is most likely to generate controversy or excitement?

Edited Note: I recognize that dropping in an "away team" is not wise or realistic, but take the question scenario as written.

Actually, I wouldn't be 'dropping in' a team of xenoanthropologists; the first step in their scientific process would be 'observation', and preferably in an environment that precludes interaction (and therefore potential contamination of the culture).

Your anthropologists, and the rest of your scientists bar a special field team which I'll get to later, should be in orbit with absolutely no contact with this race. You use drones and stealthed sensors to pick up all the information you need, and then its shared with a team up top. That team will include linguists, psychologists, economists, anthropologists, etc., all of which contribute theory in their discipline to a more complete picture of how this race lives today.

Unfortunately, these teams won't have access to the one dimension of data that we take for granted here on Earth; time. In other words, they'll see current practice, but won't know how it formed or what aspects of their culture are new, what parts come from ancient sources or practices, etc.

A really simple example of this is that many of us who are in our 50s now were raised with parents who strictly told us to 'eat everything on our plate' at mealtimes. None of us knew why at the time, but our parents were almost pathological about it. That's because they were raised by parents who had grown up during the depression, and their parents had told them to eat everything on their plate knowing it might be the last meal they get for a while. They learnt the habit because of the emotional intensity of their parents, and passed it on with some intensity, which is only now starting to die away as a parenting practice in some quarters.

So, when your anthropologists see children being chastised for what looks like a minor transgression, is that because of something in the environment now, or something that happened historically?

Enter the one field team you need; archaeologists. Your drones and satellites need to be capable of deep scans, and they need to find remote, preferably uninhabited areas where there are remains in the ground of cities, burial sites etc. that can be excavated and studied. This field work should be done without interaction with the locals, and fills that missing dimension of historical perspective for your anthropologists.

The practical upshot of all this is that the way your anthropologists would begin their studies is observation; either from an orbital platform with remote feeds, or from inside some form of duck blind, but the ONE thing they wouldn't do is interact with the culture. That would effectively contaminate the site and make many of their subsequent observations invalid.

The scientists are excited that they have a "natural experiment" and a species with ZERO shared ancestry to humans

Tim B II did a great job explaining your first question: "What is the likely order of priorities" (observation! without contaminating! and the need for archaeologists to put that observation into perspective). Much later down the line you can send in anthropologists to ask why the do certain behaviours and to run controlled studies (like they do here on earth).

I'll answer your second question "and what is most likely to generate controversy or excitement." As a biologist who studies animal and human behaviour, the answer is: you have a natural experiment with no contamination from shared ancestry, allowing you to make causal inferences about behaviour and culture.

In the sciences there are two major ways you can carry out research: observational (i.e. field research) and controlled (i.e. lab research). Field research is great because you see super real behaviours but cant control for anything (e.g. you may think island societies do x because of y but you didn't control for z which is the actual cause). Lab research is great because you control for everything but sometimes you've made it so controlled/sterile that the behaviours you are seeing aren't natural so don't actually tell you that much (e.g. you do research on parental effects of x on a behaviour but your subjects don't have access to y which is necessary for creating the natural behaviour). Having a completely separate group of animals or humans is a great "natural control" (a kind of natural experiment) meaning you can get the benefits of both field and lab research. For example you want to see the effects that living underground and eating insects has on a vertebrate. You can look at moles and say "they have super tiny eyes, so living underground must cause small eyes" but you can't prove that because what if the ancestor of moles had tiny eyes for some other reason before it moved underground, and you thats why moles have small eyes. Enter the Marsupial mole: this mole is a marsupial, so it is super unrelated to the placental moles BUT they look almost identical. Looking at both moles we can now say that evolving small eyes is probably a result living underground and not just that both moles had ancestors with small eyes (that both moles look so alike is because of convergent evolution)

To understand behaviour, "natural experiments" are great: you don't have to intervene in the subjects lives (good for ethics, and especially good in your scenario of not wanting to disturb the aliens), you get natural behaviours, you can often control for shared ancestry (yay your aliens have ZERO shared ancestry), you can often control for certain variables (e.g. you can show that smoking causes cancer by looking at countries with different smoking rates - hopefully other variables, like city density, diet, pollution levels etc, are controlled for naturally in your "experiment". Your planet likely has a lot of differences to earth (that you are now controlling for) but enough similarities to result in similar behaviours/social structures)

The behavioural biologists and anthropologists are very excited to have this other species to double check (and form new theories) on the origins of all sorts of behaviours and the effects of all sorts of environmental and social pressures on behaviour and culture. In fact every sub-field of biology is salivating at the thought of this find (and hopefully loads of other plants and animals on the planet)

Interviews with minimal impact

In addition to the great answer by Tim B II, they might want to interview individuals. Pretty much the only reasonable way to achieve that while minimizing the impact on their society and possible contamination of their practices would be the "alien abduction" sci-fi trope.

I.e. they'd want to interview a selection of people, but they'd want to do that by abducting them without anyone knowing about it, and after the interview returning them right back to where they were - after pumping them full of amnesiac drugs so that they don't know what happened and don't know what questions were asked.