How to run a Django Project on a Personal Run Server
It really depends on your requirements. Will you be accessing the website externally (making it public) or locally? Running Django from your laptop can work but if you are planning to make it public, you will need an external IP to point your domain to. Unless you have a business account, ISPs usually don't give static IPs to individual customers. Ubuntu would be a wise choice and you can run conda or virtualenv easily. 
VPS are quite cheap these days. You can look into AWS free tier that provides you with 500 hours/month on a micro server. 
If you are planning to access your website internally then you don't need anything other than your laptop or perhaps raspberry pi. If you are trying to make it available for everyone on the external network, VPS would be the best bet.
I've been wanting to run my own server for a while and I figured that running one for my django website would be a good start. What do you recommend I use for this? I've been trying to use a Ubuntu Virtual Machine to run it on one of my old laptops that I don't really use anymore until I can buy a dedicated server. 
Should I run it from a Virtual Machine? If so, would Ubuntu be best? That appears to be the case, but I want to be sure before I invest in anything. I want to be able to access the website from other computers, just like any other website. Am I going about this wrong? If so, what can you suggest me?