How can I get free users for my new app without spending on advertising?

Identify what are your values to your users. Choose the one value that you can offer without spending bucks.

Trade that, you get free users, they go home happily bringing back value earned from you.

Here are some of my 2 cents:

• Start by inviting your friends to try the app (existing contact list, email blast)
• Create incentives for existing users to share the app with their friends.
• Get a $50 Facebook advertising credit (just Google how you can get the free credit, there are many places you can get it)
• Get a $150 Google Adsense advertising credit (you just have to set up your account) and run ads up up to the budget limit.
• Share your site in Facebook Groups
• Use Q&A sites like Quora and BEAMSTART to get free SEO juice.
• Share your site on Hackernews and SubReddits.