Take it from the top
Microsoft's Office Online (known as Office Web Apps before February 2014) works with any recent version of IE, Firefox, Chrome, or Safari. It's free for personal use. Individuals and organizations with Office 365 subscriptions (from $60 to $240 per person per year) automatically get licenses to use the Office Online apps. If you're familiar with Office 2010 or Office 2013/Office 365, you should be able to jump right into the Office Online apps and get going.
While Microsoft packs a great many useful features into its Office Online suite, I was most disappointed by three problems. First, in my tests, Office Online bungled several Office documents -- real ones, gathered in the wild. The damage included a Word doc with a simple formula, an Excel spreadsheet with macros, and a large Word document consisting of photos and text boxes that, when opened in Office Online, then saved, was completely mangled. Word, Excel, and PowerPoint Online won't even open password-protected documents.
Second, the Office Online interface is still staunchly old-school Office. There's been no attempt to borrow from the excellent Office for iPad efforts. That means, among other things, you're visually bewildered by many choices that don't do much.
Third, PowerPoint Online is both buggy and severely limited. It might be useful for creating a very simple presentation or sketching out a presentation you'll flesh out later in desktop PowerPoint. But I wouldn't trust it to edit an existing presentation.
Among the three suites, only Apple's iWork for iCloud feels like it was designed, from the ground up, as a cohesive package. The result is that all the iWork for iCloud apps behave similarly -- learn to do something in Pages, and you know how to do it in Sheets and Keynote. Apple's suite is also much more finger-friendly than the other two. If you're limited to working without a mouse, there's no question that iWork for iCloud is the way to go.
The iWork for iCloud suite is still officially in "beta," which adds a note of uncertainty to this review. We don't know, at this point, if Apple will one day charge companies (or individuals) to use the final, shipping version. For now, though, it's free as a breeze. Apple used to sell iOS and OS X versions of the iWork apps, but as of September 2013, they're free for anyone who buys a new Apple computer.
My biggest disappointment with iWork for iCloud? File handling. Unlike the other two suites, iWork for iCloud doesn't have a hook into Windows Explorer. Using a file on your computer inside iWork for iCloud requires that you first upload the file into one of the iCloud bins by clicking and dragging it into a browser tab. Worse, the bins only hold files of the same type. You can't mix a word processor document with a spreadsheet in the same tab, much less the same folder.
I finally figured out how to create folders inside iCloud: You have to drag one file on top of another inside the tab. No, you can't simply drag a file from your desktop (or Windows Explorer) onto another file and create a folder. It's a rigidly two-step process.
Google Drive (Docs, Sheets, and Slides) has a feature set and interface that remind me of Office 2003. Don't get me wrong -- Office 2003 is a great product. The Google apps are free for personal use, and they come with 15GB of free Google storage. Corporate and organizational clients get socked with a $50-per-user-per-year price tag, but that brings along a bunch of management software, (very) roughly analogous to Office 365 for Business (see my review). Google Apps for Education and Google Apps for Nonprofits are free.
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