What I actually color with

If you are starting from nothing

• A dual tip water based brush pen set. This is the one thing worth buying first. The brush end fills a large shape without leaving stripes, the fine end does edges, and water based ink is far less likely to come through the page than alcohol ink. A set of ten teaches you more than a hundred cheap markers. Look at dual brush pens
• Fineliners for the small gaps. Thin, water based, and the ones coloring book artists most often name when asked what does not bleed. Good for the bits a brush tip cannot reach without going over a line. Look at fineliners
• Nothing else. Genuinely. Color a few pages before buying anything with the word "professional" on the box. You will learn what you actually reach for, and it is never what you expected.

If you prefer pencils

• Soft wax based colored pencils. Soft cores layer and blend; hard cheap cores scratch the paper and go waxy fast. You do not need 150 colors, you need a couple of dozen that layer well. Prismacolor Premier is the one most coloring book artists already own. Look at colored pencils
• If you want to go further, oil based pencils. Faber-Castell Polychromos is the one artists reach for once wax starts feeling limiting, a firmer, more precise core that holds detail better in small spaces. Worth it once you know you are staying with pencils, not a first purchase. Look at Polychromos
• A decent sharpener. The single most overlooked thing in colored pencil work. A blunt tip cannot reach into the grain of the paper, and more flat looking pages come from blunt pencils than from bad technique. Look at sharpeners
• A blending stump, eventually. Softens a join without adding color. Useful once you know what you want it to do, pointless as a first purchase. A tightly rolled piece of paper does most of the same job.

If you want the soft blended look

• Alcohol markers. They blend beautifully and they will bleed through book paper. Both of those things are true at once, and anyone telling you otherwise is selling something. Buy them knowing you will put a sheet of paper behind every page, or that you will print designs onto cardstock instead. Arteza is the accessible entry point. Look at Arteza markers
• The real middle ground, Ohuhu. Close to Copic's blending quality for a fraction of the price, the set most people graduate to once cheap markers start feeling streaky. This is genuinely where most serious colorists land. Look at Ohuhu markers
• If money is no object, Copic Sketch. The professional standard, refillable ink, replaceable nibs, built to last years rather than get replaced. Not a first purchase, or a tenth one, but worth knowing it exists. Look at Copic Sketch
• A colorless blender, usually included in a decent set. It pushes pigment around rather than adding any. Worth understanding before you buy more colors.

Paper, which matters more than pens

• Cardstock, if you print your own pages. This is the real answer to bleed. Print a design on heavy cardstock and alcohol markers stop being a problem at all. It is also why the printable editions exist. Look at cardstock
• Anything at all behind the page. An envelope. A flyer. The back of a receipt. This costs nothing and solves most of what people buy expensive markers to avoid.

More detail on all of this: which markers do what, how to stop bleed through, and how to blend pencils.