Two things to keep in mind about Newton. One is that by the time he published on gravity, he was not some nobody. He had already been the Lucasian Chair of Mathematics at Cambridge University for 15+ years, and already had a reputation as a brilliant mathematician, and was already a Fellow of the Royal Society. He was an insider's insider. The whole reason he wrote on the topic is because other eminent men pressured him into doing so, because they understood that he had worked out things that they had not and had not published on them.

Second is that he didn't just publish "findings" on gravity, but instead wrote an entire book explaining, in great detail, how to calculate gravitational attraction, and how to do it through several mathematical methods (including using calculus, which he invented). People today who have only learned that "Newton discovered gravity" know nothing of that Principia Mathematica is really about. Newton didn't "discovery gravity" — he developed a theory of universal gravitation that synthesized everything known about gravitational and orbital mechanics into a coherent, internally-consistent, highly-mathematical approach. Its advantages over anything else were pretty plain; there was nothing to compare it to at the time. He didn't just say, "there's a force called gravity," he laid out an elegant and rigorous system for understanding how all masses, from suns and planets to grains of sand and apples, are attracted to one another.

So he was definitely taken "seriously." Was he believed? Mathematically what Newton proposed was extremely useful, that much was clear, though it certainly left many unsolved problems. (The motion of the Moon, for example, was not quite accounted for perfectly by his equations — it took another century to work that out.) But "problems" are not, shall we say, a problem for new theories or world views, if they are generative problems. Truly successful new theories don't answer every possible question, but pose new questions that other scientists find interesting and worth working on. In science, a theory that actually does answer all questions quickly becomes boring (reduced to being mere "engineering," as a physicist might derogatorily put it). So Newton's theory hit that nice sweet spot of being compelling but not comprehensive, thereby opening up entirely new lines of research. Very promising indeed.

The people who took the most critical issue with Newton's theory were not fellow mathematicians, but philosophers of nature. Newton's approach relied on assuming that gravitation worked by an invisible attractive force that acted upon matter at a distance. Today we take such fields and forces for granted, but most physicists in the 17th century were still under the influence of Descartes, who, in a methodological purge, declared that all physical theories should be purely materialist, matter acting directly on matter, without any "occult forces" influencing them at a distance. The goal here was to make physical theories more materialist and thus more "scientific"; it's not a bad impulse, but it clearly went too far, as we now know. Descartes' own physical ideas were not that great and not that useful, but his methodological approach was very influential, and it is this that Newton's work most directly contradicted.

Newton did seem to be suggesting that gravity was indeed an "occult force," and this raised objections from some. Newton himself admitted that he did not know what gravity actually was, but that it acted like an occult force, and the results worked, so there. He famously declined to hypothesize about the true nature of gravity; you didn't need to understand exactly why gravity worked to see how it did. The philosophers didn't love this approach at all, but the math worked out, so everyone got over this objection relatively quickly, and tossed Cartesian physics out the window with it. (Much later, Einstein would suggest a totally different, non-force approach to gravity.)

In short: yes, he was taken very seriously, and certainly regarded as a major thinker in the 1660s-1670s. Principia cemented his reputation as one of the great thinkers of his era, though, eventually turning him into the Newton we understand him to be. Some of that mythologizing approach came later, though; Newton's reputation was large at the time, but became near divine after his death, when Enlightenment thinkers (somewhat bizarrely, because he was never like them in his thinking) took Newton as the shining example of what science could accomplish if it jettisoned irrationality, religion, superstition, and so on (Newton himself was deeply religious, and an alchemist, and so was not very much like them at all). So we can say that yes, Newton was highly respected in his time, but became even more so after his immediate time as well.

On the nature of theory adoption (and the important of posing "problems"), Kuhn's The Structure of Scientific Revolutions is the starting point. On Newton himself, Westfall's Never at Rest is the definitive biography.