I have been tasked with moving all SqlDataSource objects out of an ASP.NET pages aspx files and putting them into a separate class file but am lost. Is there a way to create a SqlDataSource object in a separate class and assign query strings to the SelectCommand, DeleteParameters, InsertParameters, etc?
To where you can call the object on a separate page rather than have the code in the aspx?
Yes, you can do that. I would recommend that you move all the database IO to a web service.
I also want to add that this is a very good step that you are taking, as far as the security of your application. Separating your data access from your user IO like this is something I consider to be a must-have security measure. Done right, you will bump up the security of your application significantly.
You can create a WCF web service easily enough. There are plenty of tutorials on the web, and I'll be happy to give you pointers as you go along. The web service would have CRUD (insert, select, update, delete) operations. You can then create an "Object" type datasource on the web page, that points to your web service, and the elements on the page can get their data from those object datasources. You can also instantiate the web service in your code behind, and use it to manipulate the data. When you create the object type data sources, you will specify the service methods that correspond to each of the commands (select, insert, update...). Hope this points you in the right direction, and feel free to ask me more in the comments, or you can initiate a chat and I will give you my email where you can holler at me.
As already mentioned, it is possible to move the SqlDataSource controls out of the WebForm. I'll assume for the moment that your current code declares these controls in the .aspx file. You could change to an imperative approach, for example, and instantiate the datasource controls in an event handler of a WebForm's code-behind class, but you don't gain much by doing so. Indeed, you could even move a lot of the code that does the instantiation into a helper class that is called by the code-behind class, but this doesn't get you much further.
The original intent of the DataSource controls was to provide developers with a way to create rapid prototypes / proofs of concept. But these controls aren't really meant for production systems. They are a violation of the separation of concerns and make unit testing difficult, if not impossible.
In some ways, the DataSource controls can be easier to work with (say, in conjunction with a GridView control). But, this convenience comes at a price, which probably helps explain (in part) why you're being asked to do something with the controls. It's unfortunate that at when ASP.Net 2.0 (WebForms) was released in 2005, the literature that was published at the time heavily promoted the use of these DataSource controls. The community has learned since then that the production value of these controls is questionable, unless you are working on simple systems that don't need to evolve much over time.
As was mentioned by Anon316, you could use a web service to handle the CRUD operations. However, this solution might not be what you really need. Additional overhead is incurred by using a web service (i.e., additional HTTP requests to the service). Having your application make direct calls to the database can still be a very good approach.
With that said, consider creating a separate class (or classes) that provide data access facilities (e.g., a Repository). Entity Framework, for example, makes creating this kind of thing fairly straightforward (and there are many other data access libraries available in the .Net ecosystem). Be prepared for adding more code to the code-behind classes of your WebForms in order to make them interact with your Repository (or other Data Access) class(es). The benefit you'll gain is more testability and reuse of your data access code. Consider putting your data access class(es) into a separate project in your solution (to start).
Whether you create separate data access class(es) in your solution or a web service, you still have significant refactoring to do in order to move away from the DataSource controls. So, again, be mindful of the additional overhead involved in using a web service, recognizing that a web service tends to make sense when you have multiple clients (e.g. web and mobile), not when you only have one.
