So next year, I'm going to be to Japan for 2 weeks. Beforehand I'll be flying to Toronto and Vancouver. But I'm purchasing all of them as separate tickets, so here's why my question comes in. I'm flying from Nashville, Tennessee, USA to Toronto, CA as a one way flight and staying for 2 days. I'm then, flying from Toronto to Vancouver and staying for 2 days as a one way flight, then from Vancouver to Tokyo as a round trip ticket. So my question is, since I'm traveling from another country (USA) to Canada, but with a one way ticket, is that okay? Will they let me out of the airport, or does immigration not allow people to leave the airport without a round trip ticket showing that you'll be leaving the country?
You don't need a return ticket, just proof of onward travel. Most commonly this is due to using a return ticket, but another ticket to another country is fine as well.
Exact requirements for a "return/onward" ticket vary depending on your citizenship and the exact circumstances. Technically for a US citizen visiting Canada, no return/onward ticket is needed, however as is always the case the Canadian Immigration official can choose to deny access to the country if they believe you will, for example, not leave the country at the end of your valid stay - and not having a return/onward ticket may lead them to believe there's a higher chance of that occurring.
However what matters is that you have an "onward" ticket out of the country. It doesn't matter where that ticket is to (although some countries will want to confirm you have valid documents, such as a visa, for your next country), as long as it's out of the country you're entering. It also doesn't matter if it's on the same "ticket", or if it's a different ticket - as long as you can show proof that you're planning to leave the country.
It's even possible for your onward ticket to be via a different method of transport. eg, you might enter the country via plane, but have a train ticket booked to leave the country.
For countries where you do require a return/onward ticket, you may be asked to provide details of it both at check-in for your flight into the country, as well as at immigration. It's a very good idea to have a printout of the details of that booking (including dates, airline, confirmation number and ticket number) in order to show at those two locations. You may not be asked to show it, but it's still a good idea to have it!