Like most developers here and in the entire world, I have been developing software systems using object-oriented programming (OOP) techniques for many years. So when I read
First of all AOP will not replace OOP. AOP extends OOP. The ideas and practices of OOP stay relevant. Having a good object design will probably make it easier to extend it with aspects.
I think the ideas that AOP brings are important. We need to work out ways to implement cross-cutting-concerns over different classes in your program without having to change the classes themselves. But I think the AOP will eventually just become part of other tools we use and not a separate tool or technique. We already see this happening.
A couple of dynamic languages like Ruby and Python have language constructs like mixins that solve the same problems. This looks a lot like AOP but is better integrated in the language.
Spring and Castle and a couple of other dependency injection framework have options to add behaviour to the classes they inject. This is a way of doing runtime-weaving and I think this has a lot of potential.
I don't think you'll have to learn a completely new paradigm to use AOP. The ideas are interesting but are slowly being absorbed by existing tools and languages. Just stay informed and try out these tools.