Do I need any tools for 3D printing other than the printer?

To print, not really, since printers come with the basics for setup and maintenance. To make prints look finished, yes. You'll want something to cut supports, trim nubs, and file or sand the rough surfaces, because none of that comes in the box.

What comes in the box with a Bambu Lab A1 or P1S?

According to Bambu's wiki, allen keys, an unclogging pin tool and a scraper blade (you print the handle yourself), plus some spares. There are no hand flush cutters.

What comes with a Prusa MK4S?

The handbook lists a uni-wrench, a spanner, a screwdriver, nose pliers, allen and Torx keys, an alcohol wipe and an acupuncture needle for the nozzle. It recommends buying cutting pliers and a plastic spatula separately.

Is a 3D printer tool kit worth buying?

It can work out cheaper than buying each tool on its own, but check what's actually in it. Many kits pad the piece count with brushes, spare needles and a second scraper you already got with the printer. Count the tools you'd use, not the pieces.

Do I need nozzle cleaning needles?

Probably not straight away. Both Bambu and Prusa include an unclogging tool or needle in the box.

What tools do you need for 3D printing?

You unbox a printer, it prints, and then you realise it came with everything you need to maintain it and nothing to finish what comes off it. The tools worth buying are mostly about the five minutes after a print finishes.

What’s already in the box

Before buying anything, check what you’ve got. According to the manufacturers:

Printer	Comes with
Bambu Lab A1	Allen keys, unclogging pin tool, scraper blade (you print the handle), spare parts
Bambu Lab P1S	Allen keys, unclogging pin tool, scraper (you print the handle), spare hotend
Prusa MK4S	Uni-wrench, spanner, screwdriver, nose pliers, allen and Torx keys, alcohol wipe, needle for the nozzle

So a scraper, allen keys and a nozzle clearing tool are covered. Loads of tool kits include all three again, which is where a lot of big piece counts come from.

What you’ll actually use

Flush cutters. The one tool almost everyone ends up buying. They cut filament cleanly (a clean square end loads much better than a bent, torn one) and split supports into pieces. How-To Geek and Snapmaker both put them top of the list, and the Prusa MK4S handbook tells you to buy cutting pliers separately.

A hobby knife with spare blades. For trimming the little nubs supports leave behind, and the odd bit of brim. Spare blades matter more than the handle. A blunt blade needs more force and slips more, which is how people cut themselves.

A deburring tool. The one people don’t know they need until they try it. The swivel blade follows an edge, so it takes elephant foot off the bottom of a print, cleans up brims, and smooths the inside edge of holes. Hackaday called a cheap one a game changer for printing.

Needle files. For flattening support scars and seams, and opening up holes that print slightly small. Get a few shapes: flat, half round and round cover most jobs.

Wet-and-dry sandpaper. PLA needs wet sanding, according to Prusa, because friction melts it into little clogged balls. A few grits from around 120 up to 400 or more covers flattening through to smooth.

Tweezers. For grabbing ooze and stringing off the nozzle before a print starts, and picking bits out of small gaps. Long curved ones keep your fingers away from the hot end.

Nice to have later

Digital calipers, once you’re printing parts that have to fit things. Bambu’s hole calibration guide uses them.
A heat gun. Prusa suggests about 200°C for one or two seconds to make light stringing disappear. Keep it moving, because PLA softens above 60°C.
Safety glasses. Prusa’s own support guide warns that support pieces can be sharp and sometimes fly off when they snap.

Kit or separate tools?

A kit can work out cheaper than buying each of these on its own, but check the contents, not the piece count. A “40 piece” kit can be 10 nozzle needles, 3 brushes and a spatula you already own. Count how many of the tools above are actually in it.

Full disclosure: I sell one of these kits. It covers the finishing tools above and has no scraper or nozzle needles, so it doesn’t double up on what came with your printer. Compare it against whatever else you’re looking at.