The T-800 says that his chip got set to "read-only" before he set off on his mission. If he can't form any new memories, he'd be about as effective a killer as Dory from Finding Nemo.
The chip isn't his memory storage; it's more like his OS and Drivers. It's set to Read-Only so that it doesn't change and adapt. He's recording and remembering things constantly, he just isn't learning from them.
This is still a bit fuzzy, as the T-800 from the first film was demonstrated as being able to process and assimilate things it learned during its time in 1985. The main difference for this film seems to be the T-800's ability to grok human morality and how it guides our decision making process, rather than simply assimilating patterns of behavior.
The reason Sarah, John and "Uncle Bob" can get supplies at "Cactus Jack's" without attracting attention or being tracked is John had just swiped $300 from the ATM before.
The T-800 says "Come with me if you want to live" when he first meets Sarah. This is likely because John knew full well that his younger self would want to save his mother, but she would be terrified of seeing a Terminator again, so during the reprogramming he set the T-800 to say the first thing that Kyle said to her, making it easier for her to believe this one was on her side.
Many viewers complained about the extra scene where the T-800 smiles awkwardly, saying that it made no sense for a machine with detailed information on human anatomy to have to scan an actual smile for reference. The T-800 has information on human anatomy, yes— but not on human emotions.
This does have a nice payoff later on during the raid at Cyberdyne, when the 800 declares that he'll handle the incoming police force, John reminds him that he promised he wouldn't kill anyone. The T-800 turns back to John; "Trust me," and gets the smile right. Which shows that the cyborg is, indeed, learning.
John Connor wears a "Public Enemy" t-shirt for most of the film. It made sense costume-wise as something a rebellious teen would wear in the early 90s, but also in the eyes of the T-1000, John Connor is "public enemy number one." Also, people familiar with the t-shirt brand would know that on the back of the shirt is Public Enemy's insignia: a silhouette with crosshairs over it. So not only is John the "public enemy" but also has a target on his back (though it's hidden under a camo shirt, and you could read into the significance of a camo shirt as well).
It doesn't seem to make a lot of sense that the T-1000 is Naked on Arrival like other time travelers even though his clothes are part of his body. There are some possible explanations though: T-1000's might be pre-programmed to mimic human skin on a cellular level but have to see particular fabrics or objects to imitate them too. Or it might imitate fabrics so well that the time machine itself would be fooled and reject it unless it made its entire body mimic human flesh.
Those mounds of human skulls that the Terminators casually crush underfoot and the treads of their tanks. How did they end up in such neat piles and where are the rest of the bodies? The people were unlikely to be standing around in tight groups when they died and, in any case, would largely have been vaporised or burnt to ashes if caught in the open. There's only one explanation - the robots collected them and piled them up. Best case scenario is that they were already dead when it happened. As to why - maybe Skynet was attempting to do a body count, or perhaps it was just an act of gloating or a warning to the survivors.
During a Deleted Scene, Miles Dyson explains his grand vision for a learning computer by painting an analogy of a sophisticated autopilot built into a commercial jet. Later, when the T-800 describes the historical events that led to the takeover of Skynet, he mentions that the first role given to the new military processors was piloting unmanned fighters and bombers. It seems Dyson's designed algorithms for a decision-making and learning processor were somewhat specialized, at least at first.