Go with Slicehost (or another VPS). Since you are a student you will be only doing yourself favours by learning how to setup and use the full stack yourself. Because it is a VPS you have to do most of it yourself and Slicehost provides nice tutorials on how to do most of it. It's been rock solid for me so far.
You can get a functional 256MB slice for $20usd a month.
Heroku is a managed solution (they do a lot of the work for you and I haven't worked out the price differences since they only just went commercial but they should be a bit more expensive than slicehost - but I am only guessing about this). Dreamhost doesn't give you as much control (but is far far cheaper).
So my vote is to skip a night of drinking a month and learn ubuntu servers setup (DNS, vhosts, apache etc, capistrano deployment, SVN and mongrel, nginx, passenger while you learn rails. Maybe also throw in SMTP, Starling & Workling and monit/god.