How much can I realistically expect to fit in my Model 3?

A bit hard to visualise "a few"... Just gotta play tetris with your bags. You can vacuum pack clothes to save space.  All the best on your new chapter!

Good idea! I guess my main concern is packing the wheels as those take up the most space.

Consider to either buy a small trailer, maybe rent or find a used one, or an overhead container you can fill. Either solution lowers your range but still you won’t speed much with all this weight so it might compensate

I’m not even sure 4 tires could fit in the trunk. I could be wrong. Never tried it.

The wheels are going to be the most difficult part. Sell them locally or pay a shipping service. The rest sounds fine; I'd just be a bit concerned leaving the car unattended for too long with your valuable inside, so do use blankets to say least cover up what doesn't fit into the trunk & frunk Consider an air pump and a tire plug kit, if you don't already have those. I got a nail on my cross country move last year and it was great being able to plug it, air it up, and continue on my way. Oh, just reread and see you mentioning taking it abroad. If you're going somewhere Tesla doesn't operate in, big repairs will be rather difficult. Even small ones can be too if Tesla does have service centers where you're going as some things aren't shared depending on where it was made. Not to mention that charge ports might differ too.

Thanks for a thorough response! Will definitely keep this in mind

My wife regularly carries 4 mounted tires in her model 3. The back seats have to be laid down but they fit. They basically take up all of the truno and most of the backseat up to the bottom of the window. You have a very few inches in front of the tires. The passenger seat, area on the floor behind the front seats and however far you are comfortable stacking stuff behind your head. I don't know what size her race tires and wheels are so the sizes may be slightly different than yours. It's a 2018 but I don't think the interior space has changed much. There's also the frunk and under-trunk area still open

Put a blanket or tarp in the back seat, the wheels will go there, with a few bags of clothes around it. Put the gaming setup in the sub trunk also cushioned with some clothes. And then everything else in the trunk and frunk

I did, it only worked with the rear seats down. The trunk alone doesn’t have a whole lot of space.

You are moving your car abroad? What is the mode of transport? I just did this for my mom. Folded the backseat down on the M3P and loaded small boxes and suitcases. But the company didn’t allow packing above the bottom window line so be mindful of that potential requirement.

Just tried it. Could only fit two in the trunk without lowering the rear seats

I will be driving myself so I must pack it in a way that allows me to safely drive it.

It will be tight with those 4 wheels but likely possible. Be sure to pack the subtrunk and frunk as tight as possible. If you can vacuum pack the clothes you can pack them tightly in those spaces to get the most use of it. Also there is a bit of space between the backseat and front seats when the back seats are down. you can slide narrow things into there before putting the seats down. The monitors might fit there and you can protect them with clothes. If you don't have a passenger you can store things on that seat/on the space in front of it too, if you move the passenger seat forward you get more space behind it, so you have a choice where to make space.

You’re moving abroad but driving yourself? I hope your windshield wipers go really fast.

Do new model 3s have towing capabilities? Shipping the wheels might be cheaper than installing a tow package.

If you didn’t buy it with a towbar, you might get it installed aftermarket. Never tried myself though, someone hopefully is more informed than me! I’d go for a roof carrier, easier to manage, more storage, doesn’t need to be removed while charging and parking

This seems cool though https://www.reddit.com/r/TeslaModelY/s/PYOSn5uvF8

A fair amount. I mean, it doesn’t have the Mary Poppins-like room of the Y, but not bad for the size.

Just use the cameras when driving and pack the back seat as much as you want..

It has a lot of space.... Just that wheels and tires ARE a lot of space.