Is Basic Mathematics by Serge Lang rigorous?

If rigor means "logical precision," the answer is that Lang is good but not perfect. He generally does a good job of honestly identifying for the reader those points where he is not being fully rigorous. These points are almost always ones where he reasonably feels full rigor isn't desirable, in view of his readership. For example, he doesn't rigorously define real numbers or the operations on them, but he does state explicitly what he assumes about them. He doesn't prove the existence of $n$th roots, but he states explicitly that he assumes this without proof.

There are some points where I would disagree with the choices he makes. For example, having accepted the existence of $n$th roots, he could have been fully rigorous in deducing the possibility of defining powers to rational exponents. Instead, he assumes that it is possible to do this in such a way that the exponent laws will work. He also never sets out what the "basic properties" of area are that he assumes as known.

But these are minor quibbles. On the whole, Lang compares well to most books at a similar level. If you don't want to wait for calculus to dot all the i's and cross all the t's, you could read The Real Number System by Olmsted in parallel.

On the question of whether Lang will prepare you adequately for Spivak, the answer is maybe. It's better than most books for this, because it does devote considerable attention to proofs. On the other hand, it doesn't develop a high level of computational skill in important areas. That's one reason why in my answer here I recommended supplementary reading. (The other reasons are to build general mathematical maturity and that the additional reading I suggested is, I feel, interesting in its own right.) The strongest students will be able to make a transition directly from Lang to Spivak. But I think many more will find they can follow the arguments in Spivak but are less successful at carrying them out themselves. The experience of reading Spivak will be less satisfying for them because they will be able to solve a smaller proportion of the problems than they otherwise would. The single most important aspect of this (after typical courses in high school algebra) is skill with inequalities, and that's why one of the books I recommended was An Introduction to Inequalities by Beckenbach and Bellman. An alternative would be a similar book by Korovkin.

With all of this being said, if you're eager to start with Spivak, there's no problem in just starting and seeing how successful you are on the exercises in the first few chapters.

Reading Spivak ought to clear up most of the concerns you may have about rigor in high school mathematics (in the sense of logical precision), but I will add a few more comments about this.

If you're exceptionally concerned about rigor, you can get the full story on the set-theoretical foundations of mathematics from a book like Introduction to Set Theory, by Jech and Hrbacek. This builds up from the axioms of set theory to a construction of the natural numbers, and later of the integers, rational numbers and real numbers. The problem is that while such a program is pre-Spivak in purely logical terms, it is post-Spivak in the demands it places on readers' mathematical maturity. For comparison, Spivak constructs the real numbers in the last part of his book, but he takes the rational numbers and their properties as intuitively known. My opinion is that few people would benefit from working through a set theory book before Spivak, but it may be helpful to have a general idea of what the steps are in providing a firm logical foundation for mathematics.

Even then, one might say that full rigor hasn't been achieved until one has presented a way to formalize the concept of proof to such an extent that the correctness of any given proof can be verified by an algorithm. (Such formal proofs are almost unreadable to humans because they are full of symbols.) This is accomplished in books on logic.

Edit: I forgot to answer the part about trigonometry. Plane trigonometry can be divided roughly into two parts: (1) geometric applications to triangles, quadrilaterals, etc.; (2) analytic trigonometry, which involves various algebraic manipulations of trigonometric functions. For the more advanced aspects of (1), good knowledge of (2) is necessary.

Part (2) is important, and I agree that Lang's book is short on this topic. But the trigonometry and complex number chapters of Parsonson's Pure Mathematics 1 and 2 (which I mentioned to you previously) ought to be quite enough for this. It's not necessary that this be taken beyond the level of Lang before you study calculus.

I think part (1), apart from the straightforward cases of solving triangles, is optional and is certainly not a prerequisite for calculus. A reasonable approach, supplementing what is done in Lang and Parsonson, would be to read some of the more geometric chapters of Parts II and III of Durell's Elementary Trigonometry and the initial chapters of his Advanced Trigonometry. (You'll be able to determine for yourself what is or isn't duplication for you when and if you study these books.) These can be downloaded here. For the later chapters especially, good knowledge of plane geometry is likely to be helpful. Kiselev (which you mentioned elsewhere that you're reading) is enough for this. An alternative could be (approximately) Chapters 10-12 of the book by Hobson, which you mentioned you're also reading.