Like others have said, the real tools for the job are a planer and a jointer. But I’ll be more specific. (And this is just what I was taught. If I’m wrong, please correct me!)
A jointer makes a surface flat. It references off the OUTFEED table of the jointer. So you press a board down on the infeed table and the blades chip away the underside of the plank. It then rides up on the outfeed table. If the jointer is adjusted correctly, the two tables will be parallel, but the outfeed will be just a bit higher, so the jointed plank rides on it smoothly. Then you press down on the board on the outfeed table and pull the board across the blades. So the start of the board might not be flat, but after it starts riding on the outfeed table, the rest of the board will be flat. (A jointer handplane was used for much the same purpose.) Very warped boards might need some auxillary support so it doesn’t rock as it goes over the blades, and maybe a few passes.
But a jointer won’t make the other side of the board parallel to the first. For that, you need a planer. It also references off the board pressed against the feed table or rollers, but it shaves wood off the TOP of the board, not the bottom. (I’ve heard of people using wide drum sanders with power feeds for the same purpose, but I don’t know anything about that.) Since the bottom surface of the board is already flat, it shouldn’t rock (though you can get “snipe” or taper at the start of the board. That means you should cut boards long before jointing and planing, and then trim.)
You can use a table saw for both of those operations, but the depth of cut is limited, obviously. You couldn’t joint a board more than 4" wide or so. Jointers can come 6", 8", 10" wide, etc.
Both jointers and planers are noisy, require a lot of space and generate a lot of chips. They can get pricey too. So some people use pre-jointed and planed wood like you’re considering, for extra money. But I’ve seen that expensive wood continue to warp and even crack after you take it home, even to a climate-controlled shop. And some lumberyards let their planer blades get dull, or they sharpen them improperly, so they leave a slightly scalloped surface on the wood, either because the wood is cut that way or because the blades compress the wood with their bevel, after the edge has cut it. Then you’ll want to dampen the wood to raise the compressed fibers, let dry, then sand. More work.
If you’re lucky, some cities have woodworker’s co-ops, where you pay an hourly fee to use their big tools. If you plan out your work, and use the place only for the tasks you need, it might not be expensive. You may need to undergo a safety training session so they know you won’t do something dumb.
Another option that nobody has mentioned yet is using engineered products like plywood and MDF. They have their weaknesses, but they are uniform and won’t warp as much (I’ve seen plywood warp, so beware.) They won’t need jointing or planing, and you can glue pieces together to create thick cross-sections. Some things become easier to build because you don’t have to account for wood movement (as much.) It can look pretty good too if you buy it veneered, or veneer it yourself. You can also apply edge-banding, but (in my opinion) the exposed plies look okay.
Good luck!