For under $100, your best bet is probably the Kindle Fire you mentioned. You can find a ton of Android tablets for under $100, but the quality can vary pretty wildly, and even one of reasonably good quality, if it's some no name brand, there is also software updates to consider. Most of those no name brands will have iffy at best support for updates.
The Kindle Fire should do pretty much everything you need, and Amazon is a trustworthy brand and should have pretty good update support. If you're a Prime member, you'll have easy access to Prime Instant Video, Prime Music, etc... Another good thing, Fire OS is Android, just with Amazon's spin on it. A pretty good portion of the apps available on Google's store are available on Amazon's too. Plus, if you're so inclined, it is possible to install some of Google's apps and apps that are on the Play Store, with a little tinkering. Fairly easy to follow tutorials can be found with a quick search.