Vulture.com did a roundup of contemporary reactions to the show this last September: What Everybody Said About The Simpsons When It First Began

In short, it was very popular, highly critically acclaimed, and somewhat controversial.

As for popularity: It was an immediate hit in the US and continued to be huge, even when it was scheduled against the well-established Cosby Show; in its first season it garnered an average audience of about 18 million viewers, which was enough to put it in the top 30 shows for that year -- a first for FOX. It was sold abroad on FOX's international stations and was equally popular in other English-language markets. The spin-off single "Do The Bartman" was a #1 hit in the UK and several other countries. Simpsons merchandise was very widely sold (this May 1990 story in People magazine suggests 15 million Simpsons T-shirts had been sold at that point), as well as knock-off Simpsons merchandise; "black Bart Simpson" T-shirts were a notable favourite.

As for critical acclaim: the Vulture roundup I linked above includes critics calling the show "wonderful", "wicked, funny, and sophisticated", "exciting", "seriously funny". Some of them express reservations, but they're pretty mild compared to the praise. Also, the show won a [Primetime Emmy in its first season] (http://www.emmys.com/awards/nominees-winners/1990?page=11).

As for the controversy: generally, it centred on Bart, who was considered to be a bad example for children, being a rude, defiant, mischievous, more-or-less unteachable, and completely unrepentant "bad kid". One of the earliest Bart Simpson T-shirts was a picture of him holding a slingshot and pointing at the viewer, saying "Hi, I'm Bart Simpson. Who the hell are you?" Another featured him saying "Underachiever -- and proud of it, man!" -- that one ruffled some feathers, with some teachers banning the shirts from their schools (though you'll note from that article that other teachers thought this was an overreaction and that Bart was no more harmful than any anti-hero character). (The show was disinclined to moralise, which was part of what educators were reacting to, but was also part of what made it so fresh and exciting by comparison to the didactic sitcoms and cartoons of the 80s. To quote the show, "There is no moral... it's just a bunch of stuff that happened.")

It wasn't just about Bart, though: the show's general air of cynicism and focus on comic dysfunction grated on some people's nerves. In this interview, Bill Cosby seems to attribute the success of The Simpsons to people being "mean-spirited". And famously in 1992, President George W. Bush gave a speech to the RNC in which he said he wanted the American family to be "a lot more like the Waltons and a lot less like the Simpsons" (YT video; also features the reaction from the Simpsons: "We're just like the Waltons! We're praying for an end to the Depression too!"). But it's worth noting that Bush lost his re-election bid, and while it would be a stretch to say that this quote was the reason, it was certainly taken as a sign that Bush was out of touch with American contemporary culture.

None of the controversies put so much as a dent in the show's popularity. If anything, they tended to discredit the critics more than the show.