As a graduate student of physics I can answer this question simply: he didn't receive much opposition when it came to special relativity. It was essentially immediately accepted.
There are a few reasons for this:
Einstein's theory was not entirely novel. It built upon the Lorentz Aether theory, borrowing the assumptions that the speed of light is always measured to be constant in any reference frame and that the laws of physics are always the same. Einstein showed that one could make the same basic predictions (e.g. Lorentz contraction) without assuming the existence of an aether.
Einstein's theory turned out to have a natural interpretation in the mathematical framework of Minkowsky 4-spacetime. It is far simpler to work with than Lorentz' theory, which had resulted in many inconsistent predictions. Special Relativity is a very neat and beautiful theory, which appeals to physicists. In Minkowski spacetime, relative time is shown to be a direct consequence of relative space; Einstein himself was surprised by this.
Einstein had a powerful logical-heuristic argument at his disposal: It made no sense that the aether acted without being acted upon. This is unlike all other systems in physics (see Newton's 3rd law). Einstein applied the same reasoning to Minkowski spacetime in his own theory, eventually leading to General Relativity. Lorentz' theory also had a similarly problematic physical explanation for its eponymous contraction (suggesting that electron orbits were squashed together for no particular reason at all). The outcome of attempts to detect Aether drag (Einstein referenced simpler first order-experiments, not the second-order M-M experiment, although it was indirectly important in that it had lead Lorentz to suggest the invariance of the speed of light) was a minor point in comparison.
The one person who insisted that SR was either incomplete, or eventually that it was the same thing as Lorentz Aether theory, was of course Hendrick Lorentz himself, who maintained that there had to be absolute time somewhere, but nobody really listened to him. However, Einstein gave him far more of the credit for the development of relativity that many do today:
The enormous significance of [Lorentz'] work consisted therein, that it forms the basis for the theory of atoms and for the general and special theories of relativity. The special theory was a more detailed expose of those concepts which are found in Lorentz's research of 1895.
Lorentz was also an early supporter for the search for GR.
The development of Special Relativity is really a prime example of a Kuhnian paradigm shift, where accumulating anomalies are eventually resolved through innovative thinking.
General Relativity was different, but not in the sense that it was opposed. It in fact received strong experimental support in the 1910s and 20s, and it solved a hundred-year old conundrum regarding gravitational lensing (Einstein describes being overwhelmed with emotion upon computing this). It was not so much opposed as ignored, in that it was so difficult to understand (requiring much very novel mathematics) few people bothered. Interest was revived with the advances in Cosmology (and, I believe, in tensor calculus) in the 1950s.
I hope this answers your question!