I agree with wat the others have said, that if you need to use someone else's computer or a shared device, QWERTY is the way to go. Unless you're one of those persons who's brain has no problems with switching, then it doesn't matter.
For me personally, although I have no experience with DVORAK, I found that a well designed, non-standard keyboard layout will make your QWERTY typing life much easier and enjoyable: the HHKB keyboard. It's very fast and easy to type on because of a few reasons:
- Caps-Lock as Control (not HHKB specific)
- Arrow/Function/Meta keys are quickly accessible from the home row
- Backspace and ESC are far easier to reach
- It simple has an awesome feel to it, typing those characters
- The small keyboard makes for very quick mouse access
It takes about 2-4 weeks to get used to (with prior QWERTY knowledge), but once your muscle memory kicks in, you'll not want to use anything else, guaranteed :) Yes, it's expensive, but the keyboard as well as keys (important!) have a very long life time, and the keys can even be replaced with a new stock set.
Besides all that, your first step is to learn QWERTY with a standard layout. I'd say, give it at least 3 months before switching to the HHKB. As for how to learn it: just try to practice 15-30 minutes a day for a month. There are a lot of apps to choose from. I used Typing Master, but other good picks are, for example, keybr.com  or 10fastfingers.com . The important thing is that you stick to the training regiment for a month. The more you train, the faster you'll pick up the basics, so you could even do it in less time. Once you got those down, practice with random texts or play typing games to further improve your speed up to a level that feels acceptable to you.
Good luck on your journey!
PS: If you're a developer and in a European country with a non-QWERTY layout, I'd recommend learning standard QWERTY and use the EurKey  layout for your language specific keys. Although make sure to know your country's standard layout at least a bit, for when you need to use someone else's machine.