Are old math books obsolete?

There are perils to reading old books. As an undergrad I looked at one edition of Maxwell's Treatise on Electricity and Magnetism, and it kept talking about "vectors", but at that time, "vector" meant "quaternion". So you have things like "the real part of a vector" and "the vector part of a vector", and it's just baffling without a lot of context. And if you tried to read about quaternions in books from the same era, you'd be even worse off. There were also all these Fractur letters for things, and frankly, many of them looked so similar that I couldn't keep them apart in my head.

There are advantages, too. Seifert and Threllfall's book on algebraic topology is a revelation, because through their "algorithmic" approach to things, you see how certain ideas (like "chain homotopy", which always seemed mysterious to me...) actually arise naturally from trying to compute things.

In general, I'd favor reading a modern book to get a sense of the modern-day notation and language, and some key ideas, and then looking at older books to see some of the things that motivate modern choices. You also get to see ideas that seemed great at the time, but didn't get much traction. And you get to see how definitions evolved as well. A quick peek at Analysis Situs won't teach you much about topology, but looking at it after a first few topology courses may teach you a lot.

The topic is very wide, but I'll try to give my opinion. I think that often it is not necessarily the maths that are obsolete, but the didactic approach of old books. Moreover, boundary breaking books were seldom made for beginners. I think that, nowadays and as a rule of thumb that might well have exceptions, the best way to start learning a new subject is to start from a beginner-friendly, modern book that puts a lot of emphasis on motivation and intuition, to get a feel of the subject, and only later try to get to the hard core, which might well be done through the original sources, when one is ready for them.

I might make an example from algebraic geometry, which is notoriously hard for beginners. Nobody would say that the best way to learn it is by going back to the original works of the Italian school and then follow the path of new developments until today. This might work, but it would take a huge lot of time. I think that a good starting point would be to spend some time on a modern book that approaches "concrete" problems, and then to get to schemes in the friendliest way possible, as they are fundamental nowadays, but also not so easy to tackle from a motivational point of view, as it is not immediately clear why they were introduced in the first place. I think that Vakil's "The rising sea - principles of algebraic geometry" does a very good job of mixing motivation, intuition and rigor. It is long but very pleasant to read and it lets you discover the important things by yourself - this is how an introductory book should feel like! Then, of course, one can still go back to older references, which are still valuable and not necessarily obsolete (Grothendieck is certainly not obsolete; and not that old, either. But the work of the Italian school might not be very rigorous, from a modern viewpoint).

As a summary and as a rule of thumb: the first books that appear about a subject are never meant to teach this subject to beginners. And, whereas some classic books are still valuable, just possibly a bit harder to tackle, for some (older) books one can really only have a (totally legitimate) historical interest, because they are very unrelated to how we do maths today.