I'm planning out a Rails app that will be hosted on Heroku and will need both geospatial and full text search capabilities.
I know that Heroku offers add-ons like WebSolr and IndexTank that sound like they can do the job, but I was wondering if this could be done in MySQL and/or PostgreSQL without having to pay for any add-ons?
Depending on the scale of your application you should be able to accomplish both FULLTEXT and SPATIAL indexes in MySQL with ease. Once your application gets massive, i.e hundreds of millions of rows with high concurrency and multiples of thousands of requests per second you might need to move to another solution for either FULLTEXT or SPATIAL queries. But, I wouldn't recommend optimize for that early on, since it can be very hard to do properly. For the foreseeable future MySQL should suffice.
You can read about spatial indexes in MySQL here. You can read about fulltext indexes in MySQL here. Finally, I would recommend taking the steps outlined here to make your schema.rb file and rake tasks work with these two index types.
I have only used MySQL for both, but my understanding is that PostgreSQL has a good geo-spatial index solution as well.