<p>I currently have a functioning in-house Windows Forms application which extensively uses the <code>DataGridView</code> control for data entry. There are some support issues which are expected when we roll this out to more locations, so one of our consultants has recommended putting together an AJAX application with substantially the same functionality.</p>

<p>I have used ASP.NET a bit in the past with the old model where basically everything goes to the server, but I've heard AJAX applications can be made more UI responsive than that. </p>

<p>So my question is, what's the best way to get started with AJAX, for someone who has worked mostly in WinForms C#?</p>

The easiest way, but not neccessarly the best way is to get familiar with the UpdatePanel, which is part of the ASP.net AJAX controls.
You can use an ASP.net GridView control (which shares much of the Winforms equiv. functionality), wrap an UpdatePanel around it and you're away.
Further on from that you will want to get involved with web/wcf services to make calls from javascript to the server and update your UI based on the responce. That of course involved more time, effort and knowledge!
There are many gridview controls you can buy that will help e.g. One from essential objects is nice, not perfect, but very cheap compared to Telerik and other big names.
ASP.net is still a server based technology, but with the inclusion of ASP.net AJAX calls to the server are easy and with jQuery updating the UI is getting easier!!
Hope this helps