There are questions asked earlier on this topic but I find that most of these books have proofs mostly. I am from software background but I can understand it if I get a sufficiently simple book containing numerous numerical examples.
Some book that patiently explains everything with simple numerical examples like say ideals on this page.
I am also aware of computational books on this topic. Tell me the simplest book with simplest examples.
Besides this book mentioned in the comments
try
Computing in Algebraic Geometry: A Quick Start using SINGULAR
A First Course in Computational Algebraic Geometry
Computations in algebraic geometry with Macaulay 2
and also