There are three popular cloud accounting systems - MYOB, Reckon and Xero. MoneyWorks is also available in the cloud - but only at a higher level (much like Reckon Accounting).
MYOB AccountRight series comes in various levels - Basic, Standard, Plus and Premier. But MYOB also offers another product - Essentials. Unlike Accountright, this does not have a desktop version. It's only in the cloud. This offers fewer features - but still does the basics.
Reckon (formerly QuickBooks), has only the Enterprise version in the cloud. This means if you want to return to the desktop, lower desktop versions won't work. But like MYOB, Reckon also offers a new cloud product - ReckonOne. Like MYOB's Essentials, this does not have a desktop version. Unlike Essentials, this comes with optional features. The core module is very cheap - but you'll probably want at least the bank feeds extra. Debtors and Creditors are also needed by most. Features can be added or dropped from month to month as needed. You may need other features (e.g. invoicing), but even with all modules it still costs roughly half of Xero's fees.
Xero is the system that pioneered the cloud accounting market (and allowed others to raise their prices). It has no desktop legacy, and the head of the company that developed it is an accountant. It had limitations in its early days - but the modern version has shaken most of these (especially in the accountant's version). It only has two versions (actually three but the limits on the first are pretty tight). Key differences are FX and payroll. If FX is a major issue for you the pricing of Xero is nothing compared with the way it handles this. Relatively new to Xero is payroll. Pricing is not cheap - but if you use Xero for accounting, it could make sense for you to use. However, others may be cheaper for you and still interface with Xero.
The other big thing setting Xero apart is third party software (like payroll just mentioned). Many extras are available - too many to even start to cover. But if Xero doesn't do what you want, extras add to the functionality but also to the cost. A common example is inventory. Programs that come from a desktop background (MYOB AccountRight and Reckon Hosted) include stock, which newer cloud-only systems don't (although there are add-ons for Xero, thus adding to the cost.)
So how do you choose between these? All have been approved by IRD (storage of accounting data offshore is generally illegal under the income tax rules). Monthly pricing is one factor - starting from $5 per month for the basic ReckonOne minimum (but probably $10 realistically) to over $150 per month for the top Xero system (prices include GST). Functions are another - if you want something particular you need to check - but as with the desktop systems, basic accounting functions are fairly standard.
Getting your hands on the system is the best way to do this. If you know AccountRight or Reckon from the desktop, then connection speed is really the only unknown. For the others, they offer a free month so you can try. Just beware that you will be charged if you go over the month.