I don't know the scale of your project, but for a smaller startup I would recommend using a hosted service such as YouTube or Vimeo to launch as self-hosted video can get very expensive quickly and isn't necessarily needed from the get go. Both APIs are fairly straightforward and there are a ton of resources out there, especially for YouTube to get you started out.
For self-hosted options you would either want to serve your videos through the cloud (AWS) or go with a service such as Wistia or Brightcove. Hosting the videos on the server you serve your site from will eventually slow you down and you probably won't be providing the best experience for your users in the long term, which is why a lot of the self-hosted options are more expensive in price.
Vimeo's pro plan has worked well for my company and allows you to whitelabel your videos which could be a good strategy to work towards starting out. They will still be served from Vimeo so you may need to consider a third-party player if that is an issue. Also, if having your domain branded URLs is a factor, you would want to consider using one of the services like Wistia from above.
Another factor to consider is that when you use a hosted service like YouTube and Vimeo you also have the option to reap the benefits of their audiences and established SERPs on top of building your own SEO (if they are public). I realize that you stated this was a subscription service, but just like Laracasts, I'd imagine that you'd probably have some teaser content to get potential users hooked into what you're selling and you'd want as many people to see that as possible!