I'd start by moving the database to it's own server first. This is the biggest bang for your buck - you may even find you can downsize the application server after moving the database out (that's true for most apps, but it depends on your app).
After you move the database to a new server, use the Percona Config Wizard to get some better settings in relation to things like InnoDB Buffer Pool.
RDS is nice too, but it will cost a lot, and the speed of queries between a DO server and AWS will create some lag as well (Unless you move the application to AWS as well).
Here's some info on using RDS with MySQL if you want to see what it looks like.
Finally, I'm working on the Scaling Laravel course which will cover moving MySQL to its own server, optimizing it, and a lot more optimizations you can look at.
Oh, and if you want to see what you can do in your application to help reduce database load, check out the first module of the Scaling Laravel course here which covers database indexes and better use of Eloquent.