The Thing's strategy is more crude in the very beginning, simply attacking humans like a wild animal. When it finally assimilates a human, it starts scheming, getting people alone before attacking, hiding its trail and trying to reason with people. Not only can it now try to replicate people, but it knows how humans behave and has learned that they can kill it.
This film is very much an Actionized Sequel to the more psychological-thriller based 1982 film, but it actually makes sense for it to be that way. Firstly, as noted above, The Thing is a case of It Can Think: it can learn more advanced and complex methods, but it needs to learn them first. This means that the Thing encountered in this film simply isn't as savvy as in the John Carpenter film; by the time of the 1982 film, it's gained the knowledge and experience to try a subtle path. Secondly, the Thing's secrecy is inherently flawed in this film; the Norwegians were the ones who dug it up in the first place. So, as soon as it escapes, they'll know it exists and is on the loose. In comparison, the Americans didn't know of the Thing's existence until they examined the ruins of the Norwegian base, so it had the opportunity to start picking off victims and replicating them before they found out it existed.
The movie happens to, by complete accident, solve at least part of the mystery of the 1982 movie's ending. The Norwegians never come up with the blood test, so the next best thing is checking for tooth fillings, piercings, or anything else artificially implanted that the Thing can't replicate. At the end of the 1982 movie, Childs noticeably still has his earring, meaning he was human.
It's actually surprisingly good at keeping the ending of the original ambiguous. Either A) Mac and Childs are both human (entirely unknown to them, but evidenced by Childs' earring - something which the Thing would probably not notice losing if it assimilated Childs, because that had to have happened outside), or B) the Thing did put the earring back in ( and see all the discussion on the WMG page about the whiskey bottle) because now it's reallygoddamn smart.
When the last iteration of the Thing chases Kate, she manages to escape by crawling into a shaft that's too narrow for the hulking creature to chase her. But as shown with Edvard-Thing, it's fully capable of detaching parts of its body to pursue victims. So why didn't it just send one of its many limbs after her?