Underlying topological spaces of scheme theoretic fibre products

Question

I am currently trying to get a basic understanding of the fibre products in the category of schemes. I am familiar with the construction done in the Hartshorne theorem 3.3. But I personally do not habe a good intuition of the construction with gluing a bunch of fibre products together ( this process is again rather tecchnical to me). My question is how to construct or obtain the underlying topological spaces of fibre products of schemes in general. To give some further motivation for the question and the answer I would like is Exercise 2.10 a) (that I have done) and Example 4.0.1 in the Hartshorne. That is for a Morphism $f \colon X \to Y$ sp($X_y$) is isomorphic to $f^{-1}(y)$, I have done this but I am not quite happy with my understanding of the reduction to affine X by gluing the spaces together in the end. The other one is the fibre product of the affine lines with doubled origins over the ground field k. It is said that the topological space is the affine plane with four origins and doubled coordinate axes, which seems reasonable, but I am not quite sure what a good way of constructing it is. I would cover the space with the affine lines and then do the gluing constructions, but to me this seems a bit inconvenient. But maybe this is what you do and you just need some experience working with this construction, I would really appreciate some personal opinions on this topic. Thanks a lot!