I highly recommend now. It's perfect to get you started, and it can be upgraded to have a more professional setup, like auto-scaling or having n instances running at a time.

zeit.co/now

It has both a command line tool and a desktop app. You can use either to deploy. It's as easy as going to the folder of your NodeJS project and running:

now

You can define which NodeJS version you want to use, it automagically will install all your project's dependencies, and will create a https URL you can visit to access your project.

Some other features:

• Domain aliasing: Besides having your domain name. It allows you to roll-back to a specific version and let's you test before you roll out a new version of your project.
• Scaling: Fixed (i.e. 3 instances) and Automatic (1 - 5 instances, depending to load) scaling for your project.
• Live logs: You can see your project's logs from the shell with now logs or in the project's URL/_logs
• DNS: Free multi-region DNS Management.
• SSL: Auto-managed HTTPS certificates.
• Secrets / Env varibables: Environment variables or shared secrets for your project (like App keys, Database settings, Provider config, etc.)

The free tier goes to sleep when it's not in use, but you can use something like Uptime Robot to keep it up (or switch to a paid plan ^_^ it's worth it).

DigitalOcean (DO) starting from just 5$/month. You can use this package even for a bit bigger production apps with small hacks and optimization. It's your own VPS, you can then install whatever you want. DO also has best and clear interface where you can manage domains and DNS records.

Sign up on DigitalOcean. It's very inexpensive if you are just getting started ($5/month). As Mev-Rael said, you get VMs. So, you can install Node.js and whatever else you need & get started!

Update: You can also sign up for AWS Free tier and see if it works out for you.