Your home router probably has no clue where that is, so it goes to its upstream router and asks if they know, this process repeats until one figures it out and you get a route.

That's not how that works. The router merely sends the packet to the next directly connected router.

Let's take a simplified example:

If you were in the middle of bumfuck nowhere, USA and wanted to send a packet to Kyouto, Japan, your router would send the packet to another router it's connected to on the west coast*. From your router's perspective, that's it; it just sends it over and never "thinks" about that packet again.
The router on the west coast receives the packet, looks at the headers, sees that its supposed to go to Japan and sends it over a link to Hawaii.
The router in Hawaii again looks at the packet, sees that it's supposed to go to Japan and sends it over its link to Toukyou.
The router in Toukyou then sends it over its link to Kyouto and it'll be locally routed further to the exact host from there but you get the idea.

This is generally how IP routing works; always one hop to the next.

What I haven't explained is how your router knows that it can reach Kyouto via the west coast or how the west coast knows that it can reach Kyouto via Hawaii.
This is where routing protocols come in. You can look up how exactly these work in detail but what's important is their purpose: Build a "map" of the internet which you can look at to tell which way to send a packet at each intersection depending on its destination.

In operation, each router then simply looks at the one intersection it represents on the "map" and can then decide which way (link) to send each individual packet over.
The "map" (routing table) is continuously updated as conditions change.

Never at any point do routers establish a fixed route from one point to another or anything resembling a connection; the internet protocol is explicitly connectionless.

* in reality, there will be a few local routers between the gateway router sitting in your home and the big router that has a big link to the west coast

That sounds like quite a messy and inefficient process! But I guess as long as it can be done quickly enough, it doesn't really matter?

Every time

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Oh wow, this unlocked a memory! Pretty sure I watched back in school. Quite informative, though it felt like it skipped a lot between leaving the host computer and reaching the destination - is it just the same process over and over until it reaches the right place?

I remember watching this in high school IT class lol

To expand on this: every website that you go to online (i.e. www.google.com) is backed by an IP address.

it doesn't send it down the correct cable. It sends it on.

Imagine your friends. you need to talk to somebody. Lets call him Garry. You don't know Garry's contact info. So instead, you pull out your phone, and text Sally, asking her to ask Garry if he knows where your glasses are. Sally pretty much knows every one. Or at least, you thought she did. Reality is she sent to to Becky who sends it on to Steve. Now, Steve is the one who invited Becky to Garry's party, and because... reasons, Becky invited Sally who invited you... so now, Steve relays the question to Garry.

Garry hasn't seen your glasses, but, he does have a weird set of car keys with a giant Charzard key fob... maybe they're yours? So, he sends his reply to Steve, which forwards it to Becky, who sends it to Sally, who giggles and asks if you really have a charzard key fob.

You get the idea. Only unlike people, the data usually doesn't get mangled.

I wrote up a whole thing that didn't post. There's good answers here but I think that, like me, you wanted a more "voltage based" one.

Short answer is they don't. Everything on the network is always listening, and security is based solely off of a handshake. Everything is always employing a fancy multimeter that measures voltage high/low as a 1/0 turning it from bits to bytes etc. The router listens to that and decides where to send it upstream, which it isolates from downstream.

For a realllllly basic example look at the modbus protocol. That's also why industrial equipment folks get real touchy about network access. For things like computers, theres talk back and forth to verify. Modbus is just "if the byte is the thing I do the thing". But fundamentally, that's the physical basis: all devices are always listening, the TCP/IP stack is what tells them what to disregard.

Ethernet is the common PHY even across fiber afaik.

The circuitry doesn't determine which cable is the correct one. That is determined by a protocol that associates various IP networks with different network interfaces. So, for example, all data going to 192.168.5.0/24 goes to interface eth0, and 192.168.0.0/24 goes to eth1 and 10.0.0.1 goes to eth2 and so on. Each interface is a separate RJ45 Ethernet port on your router, for example. It doesn't have to be RJ45 it could be your router has a Thick Ethernet or Thin Ethernet connector. Or it could have wifi. Or something else.

Anyway, forwarding the packet to the correct interface / subnet can be done with a static route defined on the router. Another way is dynamic routing using BGP (border gateway protocol) which is an exterior gateway protocol that dynamically routes between your network and somewhere exterior to your network. Yet another protocol is OSPF (open shortest path first) which is used inside a corporate network for dynamic routing.

For any of these the router knows how to send the IP packet to the next hop, another router, which in turn knows how to send it to the next hop.

Where to send is based on the destination IP. The routers know which interfaces and which other routers are responsible for different subnetworks.

It is sort of like how once your mail makes it to a main hub in your state, it is then routed to the main hub for the destination state, and from there to the post office responsible for the destination zip code, and then to the mail route (and hence truck) responsible for the street and number.

So if your destination is 1.1.1.1 maybe there is a router known to be responsible for 1.0.0.0/8 and then it knows what router is responsible for 1.1.0.0/16 and so on until we get to a router that has 1.1.1.1 on one of its subnets then it sends directly to 1.1.1.1.

Look into MII, RMII and RGMII if you want a google starting point

Thanks, this is a good summary. It's useful to know about the dynamically changing route - that explains a lot.