Web app deployment process and tools

This is not a technical question but rather one about the process of pushing an update from development to the live environment.

I'm working on a web application which is scaling and currently lives on a shared hosting account where I basically upload my new files via FTP (filezilla) if I have do an update. Due to the increasing load and traffic I need to sort a few things which are listed below:

1: Hosting / managing server

Because we are scaling i'm facing the choice between going on a VPS (€ 15 / month) and managing the server or using a PaaS like heroku (€ 75 / month).

2: Deployment process

Many times when I update I'm scared that devices using the webapp will face an error while updating and that users will have down time. To eliminate any bugs I'm incorporating E2E and unit testing. I see tools such as codeship, circleci, travis ci, jenkins,...

What I would like to know

1. How does it all fit together? (Github / Heroku / CI) - What is the flow?
2. VPS vs heroku, I'm not sure what to choose here...
3. Any other tips about pushing updates to live environment?