Update: After episode 5, still loving the show.
I'm writing this after the first two episodes. There are some things that are still unclear, so I shall be updating this as new episodes come out.
I loved Clone Wars and Rebels and think so far this is doing great at bringing these characters to live action. I think they did a good job at addressing points from the animated shows that are need to know, so if you haven't seen them you should still be able to understand Ahsoka.
Obviously the characters aren't exactly the same as they were in the other shows. But this is set 10 years after Rebels, and 30 years after the Clone Wars. It would be weird if Ahsoka acted the same way she did at age 13. She behaves in this much like she did in the Mandalorian, and I feel that her portrayal in Rebels bridges the two well. One thing her more serious attitude now can be attributed to is that she only found out near the end of her time on Rebels that Anakin had turned to the darkside, and she now feels somewhat responsible because she left.
I have read some complaints about Sabine being depicted as moody in this when she wasn't in Rebels, and I'm shocked by that because I had the opposite reaction. I felt she was sometimes moody in Rebels (not all the time, I did always like her character) and in Ahsoka she is less so. I really like this depiction of Sabine 10 years older, and look forward to finding out more. I wasn't expecting them to make her a Jedi, but I feel like the way they're doing it respects the existing canon and I trust Filoni to do it well.
Finally, I want to clarify a timeline confusion that I've seen in some of the reviews here; people are talking about the fact that the end scene of Rebels set up Ahsoka and Sabine looking for Ezra but now it seems like they are starting again. The answer to this is that the end scene of Rebels isn't set before Ahsoka, it is in fact recreated near the end of the second episode of Ahsoka. So the search they are starting now is the same search that we saw them start at the end of Rebels, we just have more context now.