Let $X$ be a projective variety embedded in a projective space, over a field of arbitrary characteristic.
What is a good reference for a nice proof of the classic fact that degree of $X$ is well-defined, that is, a "general" plane of complementary dimension intersects $X$ in a fixed number of points? (assuming Bertini's theorem). A proof which requires less amount of background is preferable.
See Fulton, "Intersection Theory", Ex. 8.4.12, p. 149.
As Kaveh says, the best way is to follow Hartshorne's Chapter I, Section 7. This is before he gets to schemes, so it should be considered "classical".
Thm 7.7 in that section is a slightly generalized version of Bezout's (classical) theorem on the number of points plane curves intersect in. It considers intersections with hypersurfaces. Now let $X\subseteq \mathbb P^n$ and its degree defined via its Hilbert polynomial (as on page 52). Then by (the proof of) Bertini's theorem, for a general hyperplane $H\subseteq \mathbb P^n$, $X\cap H$ is reduced and then by Thm. 7.7, $\deg (X\cap H)=\deg X$. Keep cutting with hyperplanes until you get to finitely many points and it is easy to check that the degree of a reduced zero-dimensional scheme is the same as the number of points in it.
I suppose one of the best approaches (In my humble view, the best one) to delve into the concept of the degree of a projective variety is the method that has been used in "Algebraic Geometry" by Robin Hartshorne.
More precisely, if you study Chapter 1, Section 7 (Intersections in Projective Space), it will give you a real insight into the matter. According to the fact that, as it is mentioned in the aforementioned part of the book, a purely algebraic definition of degree has been given, it would be much easier (at least to me) to gain a clear answer to your question.
I would like to mention that Theorem 7.5 (Hilbert–Serre) plays a vital role in the aforesaid section and if you are already familiar with the Hilbert polynomial of a graded module over a polynomial ring and its Hilbert function, you can directly go to Hilbert–Serre Theorem and the rest of the section. Otherwise have a quick look at the whole section and in a few minutes you will get what you want.
I hope it will help.
Edit (May 12, 2022): While it is probably too late for the purpose of the original posting, but I also wanted a more "direct" way (compared to the approach by Hilbert polynomial) to see degree as the number of points in the intersection with generic linear subspaces with complementary dimension. I could not find a reference, and eventually ended up writing it up myself (see Example III.122 and Theorem III.123 in Mondal - How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)). It does not use Bertini's theorem, so in positive characteristic needs to count points of intersection with appropriate multiplicities. I am happy to report that while I had forgotten about this question, I actually ended up doing precisely what I suggested 10 years ago, namely following Mumford's proof from "Algebraic Geometry I".
This does not answer your question, since the proof works only when the field is $\mathbb{C}$, but when I was learning it, the most accessible and understandable proof for this case was that of Theorem 5.1 of Mumford's "Algebraic Geometry I: Complex Projective Varieties".
