In Stranger Things, what kind of research exactly was being done at the lab in season 1?
Research on psychic powers and their effects
I will address not just Season 1, but rather the whole program, including that indicated in later seasons.
The chronologically earliest part of the program seemed to involve doing experiments on people who either initially had psychic powers, or who developed them. For instance, the mother of Eleven either developed powers due to experimentation during her pregnancy or already had them; in either case, the experiments on her seemed to be focused on her child: either simply waiting for them to be born to take them away, or perhaps actively trying to make them a (more) powerful psychic.
We also see this with
who seems to have been taken to the lab as a fairly young child.
This is because, contrary to the general idea in SFF works, disaffected adults with a grudge against the system who have been horrifically mistreated in experiments are not the best candidates for research seeking to bestow weapons-grade supernatural powers—at least not for the people doing those experiments or the organization funding them. It is a better strategy to work with children or even babies raised and indoctrinated to believe in the system, although if you are really committed to the horrific mistreatment part, this rarely works out much better.
Once they had their theoretically compliant test subjects, of whom we have seen two (or looking at it differently, zero) the primary goal was to get an advantage over the Soviets with their new super soldiers by teaching them to develop their powers through Pavlovian reinforcement and extraordinary disregard for their well-being. For instance, in Eleven's case they were clearly trying to get her to spy on Soviet officials. This mirrors the real-life research into remote viewing during the Cold War, which unsurprisingly and as far as we know, failed to yield any interesting results. In point of fact, the government correctly recognized that Eleven would be far, far more valuable as an intelligence operative or asset than slinging around Soviet soldiers with telekinesis. In the case of
the goal was presumably to have her use her misdirection abilities in targeted attacks against foreign targets. Probably assassinations. That said, I am sure that battlefield deployment was certainly an option that was being considered for extreme circumstances.
The creation of, or perhaps merely access to, the Upside-Down was a direct result of...something that happened during one of Eleven's remote viewing sessions. It was not, so far as we know at this point, an intentional result of the initial research program. She punched a hole in the wall between the two dimensions purely by accident.
However, once they had access to this golden source of potential advantages over the Soviet Union, they were hardly going to simply seal it off and do nothing. So the focus of their research then shifted toward utilizing and exploiting this dimension. I am sure they believed that it was the secret to being able to drop a strike team on the Soviet Premier without any warning, or bioweapons that could be used against their foes. And they were right...too right.