The Wights have no free will. Whether they also lose their memories is never explicitly stated, but we can try a guess (primarly based on the books);
Othor was an experimented ranger of Castle Black during his life, he knew that the most important man there was the Lord Commander and where to find him. As a Wight, he tried to kill the Lord Commander. It seems unlikely that he just randomly wandered there; it was at night and all the doors were closed, but he ignored everything else and went straight to the quarters to the top of the "Commander's Keep" Tower, killing only the guards on his path while the other one simply attacked anyone coming close to him (depicted in episode S01E08).
It's also the opinion of the Lord Commander Jeor Mormont:
"One slew Ser Jaremy Rykker and the other came for me, which tells me that they remember some of what they knew when they lived, but there was no human mercy left in them."
A Clash of Kings - chapter 23 (Jon III)
Now, as stated by @kuhl, he could have been remotely directed there. But according to the legends, the Wall has been raised after the Long Night (the war against the White Walkers), and there are "great spells locked beneath its ice":
- We saw it blocking long-distance magical abilities; Jon Snow (who is a skinchanger in the books) can't sense the direwolves on the other sides of the Wall (first Ghost after he climbed the Wall with the Wildings, then Bran's direwolf).
- Once Jon became Lord Commander, he wanted to experiment with the Wights. So he put some bodies in the Ice Cells (storeroom carved inside the Wall), but nothing happened.
So, does the Wall completely inhibit magic? Apparently not. Othor and Jafer Flowers were already Wights when their bodies were found north of the Wall (Dywen: "And might be I'm a fool, but I don't know that Othor never had no blue eyes afore" - ser Jaremy Rykker: "Neither did Flowers"). They were brought back to Castle Black and put into the Ice Cells so that they don't rot before the "autopsy" by Maester Aemon. But unlike Jon's experiment, that didn't stop them (but in their case, they were already Wights before that).
We don't know exactly what kind of magic the White Walkers use to raise the Wights. It could be a variant of skinchanging, where they would take full control of the bodies in a conscious way. But:
- There are a lot more of Wights than White Walkers. If all of their movements were to be consciously made by White Walkers, it would be very hard to manage more than a handful of them at the same time
- We never saw a dead body staying dead north of the Wall. They seem to rise as Wights even without a direct touch or sight of a White Walker: Torwynd (Tormund's son) died of cold/sickness at night, shortly after the Battle of Castle Black (S04E09 of the show) in a tent with his father and rised as Wight before anybody noticed he was dead. Given the habbit of the Wildings to have fires and sentinels all around their camps, it seems hard for a White Walker to have unknowingly approached enough to be able to reach him or know he was dead
Knowing how to fight is one thing, but you still need strength to be a good fighter. On that aspect, we can already tell that the books and the show disagree:
- In the books, the bodies don't magically move. Their muscles can still act even without a nervous system (Jafer Flowers managed to kill ser Jaremy Rykker after having lost his head, and there are numerous examples of cut limbs still moving, Pirates of the Carribean style). But the more the body is damaged, the slower and clumsier it becomes, until the point where it can't move at all once it's completely rotten. From that, we can suppose that the stronger the man was, the stronger the Wight is.
- In the show, we have seen corpses that were hardly more than skeletons (in episodes S04E10 and S05E08) who were still fighting and actually quite fast.
So, my guesses in short:
Books Wights: closely tied to the capacities of the living human, but quickly decreasing.
Show Wights: don't know, but probably disconnected.