You don't. You have to use iTunes to add music to the bundled Music app. You can use other music apps that have different ways of getting the music onto your device, e.g. Spotify, last.fm, Amazon Cloud Player, etc.
Just like any other mobile device in existence since the invention of Plug and Play and USB.
You don't like how the bundled Music app works but you don't want to install a different music app? You're setting up a no-win situation there.
You can do it, but it's in no way straightforward: all sort of cloud services (no, I don't want to send my files over to your servers in Finland to copy things in my living room), making the iPad a WebDAV server (probably the simplest solution but definitely annoying), SMB shares (only seen on the iPad through crappy or paid apps), iTunes (again, crappy solution, a lot more clicks than a real file manager).
I mean, just because Apple (or the media companies Apple does business with) has an agenda to push you can't blame me for bitching at them. After all drag'n'drop file copying between devices was a problem solved 20 years ago. Now I need special "apps" on either my PC or on the iPad (most of them crapware or paid or paid crapware) to connect glorified "iDevices". The just took that option out - there was no technical reason to not have it and just HIDE the entry from newbies.
Add to this the fact that the app market looks so immature that I feel I'm back to DOS shareware land. Few major companies I feel I can trust to not screw me at a low level (I don't care about NSA intrusions, I care about credit cards numbers being leaked out or major security breaches on code no-one has ever reviewed), no well known, mature OSS apps for the main things I do with my device.
Right now installing an app seems a crap shoot.
Anyway, it's all business reasons and not tech reasons, I get it. End of rant :|
When I use Android phones I have literally dozens of options to get music to play with the stock player. (copy it from a cloud storage, adb push it to my phone, copy with a simple file explorer, download it from a website or use btsync to sync it automatically)
Apple made a big step when the iPhone/iPad no longer needed a PC to just activate it. And then again a big step to allow OTA updates. I think the next step would be to allow some way to add music without using a dedicated PC.