A general pitfall in numerical methods is settling on a bad old method without understanding what all your options are. It's easy to get stuck on suboptimal methods, especially if you don't look at more recent references and the literature. Not all new methods are good; lots of old methods are great. There are a dozen different ways to solve any given problem. Usually, each method will be ~equally as hard for you the implementer to understand, but they can have wildly different performance characteristics. This is the main reason to avoid looking at something like Numerical Recipes---it will get you stuck in a local minimum.

I'll try to give books that are good starting points and provide a useful and correct orientation for each topic. For each of these topics, there are many, many other books. I don't know all the books---I'm sure I'm missing other great references.

General numerical analysis reference: Suli & Mayers, Stoer & Bulirsch, Ridgway Scott's book. There are many others---these are the ones I like. Suli & Mayers is truly excellent. Stoer & Bulirsch is comprehensive.

Linear algebra, etc: Trefethen & Bau, "Matrix Computations". Yousef Saad's books. Greenbaum's book. There are other key topics not covered much by these, namely sparse direct solvers. Tim Davis has a good intro book. Martinsson has a new book on fast direct solvers---unclear how useful such solvers are in general.

Approximation theory: "Approximation Theory and Approximation Practice" by Trefethen. Extremely important book. Many numerical methods for solving 1D problems (rootfinding, approximation, optimization, solving differential equations, nonlinear boundary value problems, representing special functions, etc) are made completely obsolete by using Chebyshev approximation, at least when the function you're working with is analytic (and even when not, you can get very far by using a "Chebyshev spline"). If all you're doing is calculus on a computer, this is probably the only book you need to read.

Optimization: Nocedal & Wright. Boyd & Vandenberghe. Bertsekas has great books.

Interval arithmetic: Moore, Kearfott & Cloud. This topic doesn't get as much play in the academic applied math/scientific computing/numerical analysis community, which is a real shame, IMO. Gives you the basic tools to get into verified numerics, where you can write algorithms which work provably correctly, allowing you to avoid epsilon hacking to some extent. E.g., "do X if |small number| < eps, otherwise do Y". Without verified numerics it's impossible to build correct algorithms on these kinds of predicates with floating point arithmetic).

This covers what I would consider the "basic" topics. At least, if you're doing something else that's higher level (like solving PDEs), you need stuff from this toolbox.