So a buddy and I do php development at work and we have a dev and live server, but they are mapped directly to our computers at work so we can just open up the files and edit them both on the dev and live servers.

That is what we are used to.

Anyways we want to develop a project in php on our own in our spare time. How would we go about setting up a dev / live server similar to what we have at work or a better way?

Cheapest is just to set up a LAMP server at home on an old desktop or laptop.
http://lamphowto.com/

What's an alternate solution? We don't have any extra PC's that can be used as servers.

Use cloud hosting? Can probably get away with using AWS free tier depending on your requirements.

https://www.paperspace.com/

You can probably try that if you need better specs than from the AWS free vm

You can get AWS for free for a year. And after that if you use a micro instance with no load balancer, it costs like $10 / month if you keep it running 24/7. If you shut it down for the night and work hours (assuming you only do your project in the evening) you could have it for just a couple bucks a month. Essentially free at that point.

Xampp works great and you can setup TeamViewer to remote in. You can also purchase a domain name if you don't want to type in your main ip to your browser each time. It's free and pretty simple to setup runs on any windows pc so you dont have to dedicate one for it.

Get a Raspberry Pi 3 and set it up as a LAMP stack headless server
It's like 36$

Use a program like xampp to test all your code locally. Then push it to the live server. The live server would be something like LAMP.

Just get a cheap vps for like 2 bucks a month.

Setup httpd with it, then I'm guessing Ur using Dreamweaver. Connect to the server through Dreamweaver. Then u can sync it automatically with it