I would like to understand what other ways I can use Bluetooth on my iPad apart from connecting a physical keyboard.
You can do peer-to-peer networking on bluetooth with some games and other applications.
I haven't checked out the details but my understanding is that the Bluetooth profile is much better in the iPhone 4 and iPad than in the early iPhones. I expect you could also hook up to BT headphones and speakers to stream music.
In addition to P2P sharing and audio streaming like Doug said, if you have a jailbroken 3G one, you can create a wifi hotspot using MiFi.
Also, some emulators allow you to use a Wii controller via Bluetooth.
There are Bluetooth-to-IR devices (for example) that let you control your TV and other devices from your iPad over Bluetooth.
You can “tether” your iPad to an iPhone to use its cellular data service as a “mobile hotspot”. For when you need internet access on-the-go.
For details, see MacWorld article Use Bluetooth to tether your iPhone, iPod touch, or iPad by Glenn Fleishman.
By the way, a third way to tether is via USB cable.
Starting with iOS 7, Bluetooth on iPad (as well as other select iOS and Mac devices) can be used to exchange files and information wirelessly in an ad-hoc manner using a feature called AirDrop.
Bluetooth is also employed when setting up and communicating with other Apple devices like AirPods, HomePod and Apple TV. With Bluetooth turned on, even a new iOS device can be easily set up by transferring content using an iOS feature called Quick Start.
Another interesting application with Bluetooth turned on is to easily share Wi-Fi password with nearby iOS devices belonging to saved contacts without entering it manually. Instructions can be found in the 9to5Mac article:
iOS 11 includes many automatic features regarding first-time setup and account login. This extends to WiFi networks too. If a friend comes over with an iPhone running iOS 11, you can automatically log them in to your WiFi in one tap.