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Beyond the Core Rukebook, the Kill Team, their rules, a stack of dice, a tape measure and the tokens that come with the Kill Team, what else do I need to play?

Books/suplements? None. Team rules are available in the free KT app and in a printable version on Warhammer Community.

You don't even need the Core book. Quite a few of the printed rules are already outdated. The official app -annoyingly- only has a summary of most basic rules. You're much better off using online resources like https://kt3.albecortes.com/ or https://wahapedia.ru/kill-team3/the-rules/core-rules/ as they are up to date and even provide information on which parts have been changed and link to relevant erratas and FAQs.

What you will need, though, are generic tokens, as I believe the ones in team boxes are only the ones to mark the abilities of the team in question. In particular Engage/Conceal tokens as they are key to gameplay. Once you know the game better you will see what other markers you may need - for example certain equipment options. But you just can't play without Engage/Conceal tokens (you may make your own, though, or buy them in many varieties from online sellers).

A couple of D8's and D12's to mark how many wounds wounded operatives have left wouldn't hurt either (D10's aren't that stable, that's why I recommend 8/12's)

I'd check out Kill Team Dash (https://ktdash.app/) great source of information and I find it very helpful to track abilities and other things as I play.

You could also get a laser pointer for line of sight. I like the Kill Team: Approved Operations Card Packs since it has the missions. Also, the equipment pack if you plan on showing what equipment you're taking.

Oh, yes, a laser pointer is very useful in KT.

You only need your models, engaged/concealed/wounded/+1AP/-1AP tokens and the rules to play. The various missions for each campaign can be found in the associated campaign Dossier. I also recommend getting some tiny die for wound tracking (I use tiny D6s myself, some people prefer D10s, use what fancies you).

A laser pointer is also useful to have around, as already mentioned, but by no means essential

If you stick to playing Solo for the first few games, you don't even need an opponent. In fact, playing co-op games is a great way to teach new players the rules of the game

Ah yes, the Crit Ops card pack is probably the cheapest way to get a bunch of missions. They aren't super flavourful though