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I'll give you some practical suggestions. Start with a box knife or xacto knife. You'll want several blades because the sharpness really matters to get a clean cut. You'll also want a metal ruler for measuring and to help cut straight lines. But even better IMO is to add a paper trimmer and break off the blade track from the bottom. It won't cut all the way through but it's way easier to cut straight pieces if you use it to do an initial cut.

Next, get some pins with plastic caps on the end (so you can pull them out). In addition to holding pieces together while they dry, I use them to test everything for fit before I commit to gluing. The glue I use it just some craft glue.

I got all of this and the foam board in one trip to Walmart. It's super easy to get into.

One more piece of advice: it might be attractive to make your first insert for the game that gives you the most organizational trouble. But you should really start small. You'll learn a lot your first few times and making a big game with a lot of pieces fit back in the box with an insert can be hard. I did Merchants and Marauders first. I forgot to leave room for the board so now it doesn't close all the way. Whoops.

I've become a bit of a foamcore junkie, and my designs have gotten much more complex as time has gone on. I keep meaning to post here, but always forget until my design philosophy changes again and I think "I need to wait until I get more pictures of X", then the cycle repeats.

Part of my evolution has been changing how I do joints. I want to start off by saying this -- unless you are a perpetually unsatisfied perfectionist like me, pinning and gluing is more than enough. In fact, I rarely find pinning necessary if you have a good triangle set in several sizes.

For the love of God, don't dove-tail. It is not worth the time you will spend on it and you will hate yourself after a while. I'm not saying this from direct experience, but given the experimenting I have done, just the thought of cutting dovetails makes me break out in hives.

The middle ground that I have found in both increasing attachment surface while maintaining efficiency is making rabbet cuts with the FoamWerks rabbet tool. Rabbets cut part of the foam board away, leaving an extension that consists of one papered side and a small amount of actual foam. You can glue each joint on two surfaces, so it's much stronger, and has the side benefit of hiding foam edges on all but the top surface.

The rabbet is not perfectly calibrated, so there is always 1-2 mm of excess that needs to be trimmed away, and it changes the way pieces fit together by a few mm, but once you get used to that the process of rabbet cutting your joints only takes an extra minute or so per joint. To me, the aesthetics of it make it worth doing.

ALL OF THAT said, if you just want something functional, and are not using it as some kind of twisted art/engineering project, then square cutting and gluing joints is perfectly fine.

This is what I have been using and it is great - perfect 90 degree corners and straight lines

https://www.amazon.com/dp/B005XUHIBG/ref=cm_sw_r_cp_api_Q0YfzbMQW2PYX

Not cheaper bit I think better.
The Logan stright cutter
I use it with my mat cutter but it works with a good metal ruler as well.

I used this. Took a few tries on scrap foamcore to get the hang of it, but really happy in the end.

OLFA 9911 CMP-1 Compass Circle Cutter https://www.amazon.com/dp/B000BK7NWC/ref=cm_sw_r_cp_api_i_n3nwDbMEMY3MJ

This is what I use as it was suggested here: Aleene's Quick Dry Tacky Glue 8oz https://www.amazon.com/dp/B001145496/ref=cm_sw_r_cp_apap_rq8W2KUeR31EB

You don't need the quick dry and some people suggest not using the quick dry because without it it gives you more time to fix mistakes or reposition pieces. That's probably true and I have made a few mistakes that I'm sure would look better and been easier to fix if I wasn't using the quick dry but I do like the speed.

I use a basic retractable utility knife and stand directly over the foamcore which makes it pretty easy to keep the blade straight up over the cut I'm making and deal with the gap. Your mileage may vary though.

My non-slip ruler is a T-square but the square doesn't seem to be 100% perpendicular, so I'm going to return it and just get a basic non-slip ruler. I've had good results just by measuring a couple points, marking them with a pen and cutting from point to point.

Here's some that I bought. Admittedly, I have built anything, yet, but it seems like it will work well.

I use this with a metal ruler and a scrap piece of foamboard underneath. Works pretty good and can even do bevel cuts!

I use Olfa craft knives and blades, I use the super sharp blades. 50 pack, 13 segments per blade, they last for quite a while.

https://www.amazon.com/OLFA-9149-ABB-50B-UltraSharp-Snap-Off/dp/B0006SJAQ6/

That's 9mm, I just picked up an 18mm version as well for my next attempt. The 9mm blade can get a little wobbly on long cuts.

Aluminum is softer than steel and you are likely to scrape some off every cut, making the edge untrue after a while (but not as quickly as plastic of course)

I would recommend stainless steel with a non-slip cork back, about 18" in length, with both inches and centimeters. Example

The T-square will be good to get straight cuts. They come in steel too, but are more expensive. I guess if it's only used occasionally, and carefully, the aluminum model might be fine.

You can not use a paper cutter but if you want to use a Mat cutter that works. But they are fairly expensive. I actually have one and use it the Logan Artist Elite
https://www.amazon.com/gp/product/B004V8YA5S/ref=oh_aui_detailpage_o04_s00?ie=UTF8&psc=1

Foamwerks hole drill

I have an Xacto circle cutter.... https://www.amazon.com/X-ACTO-X2020-Designer-Circle-Cutter/dp/B004DNX71E

I switched to this glue which dries fast enough that you only have to hold it on for 30 seconds or so before letting it sit (though don't bump it afterwards for another hour or so). Since I switched, I haven't needed to use pins at all.

Maybe a leather punch? Hit it with a hammer and it'll pop out a circle.

https://www.amazon.ca/Hollow-Punch-Leatherworking-Punching-Craft/dp/B000RB1K4C

Foam cutting hot wire table.

https://www.amazon.com/Proxxon-37080-Wire-Cutter-THERMOCUT/dp/B0017NS8H6