Aditya Jayaprakash

Congrats on your side project.

Since you emphasize maximizing the number of free users while keeping costs low, here is the tech stack I recommend.

Building your app

Frontend - Vercel. Vercel is free if it's a hobby project and easy to start and manage.
DB - Neon/CockroachDB. Neon would be an excellent choice for your use case since you're already on Postgres. Both are serverless and offer very generous free tiers.
Server. There are several options I would recommend - Fly.io, Railway and Render. I've used Fly in the past - it's easy to use, and you can initially start with machines that cost < $5/month and scale up as you see fit. Render offers a free plan for hobby projects.
Auth. I would strongly recommend not building your own auth system and instead using Clerk or Auth0 or Firebase or Supabase. I've used Clerk, and I can't recommend it enough. I've also heard great things about Supabase.

Landing page - Typedream or Framer. Their starter plans are less than $10/month and let you create a high-quality landing page with little effort.

Unless you're already familiar with or intend to get acquainted with large cloud providers, I would not recommend hosting this on AWS/GCP or Azure - to minimize setup time. Microsoft does offer a very generous starter plan for founders. +1 to what Casey and Rahul mentioned about using frameworks that have already been battle tested and have plenty of documentation available.

Distribution

I would primarily use the following to broaden your reach.

Social media - Twitter/Facebook/LinkedIn. This works great, especially if you have an audience already. Alternatively, you could ask your friends to promote it.
Content marketing. Another way is to create content relevant to what your tool and popularize your tool through it. YouTube is an excellent choice to plug your tool into your video. Another way is to write blog posts relevant to the problem you're solving. You could also post this article on relevant subreddits.
Do a Producthunt launch.
Showcase your tool on Hackernews.
There are several tool aggregator sites. Here's one example of an AI tools aggregator. You could find a similar site relevant to your tool and post it there.
SEO - It's free and could be a great traffic source. The downside of SEO is it's a slow process and takes between 6-12 months to become fully effective.

Happy to talk and answer more on this topic.