Michael,
The answer depends on the tasks, you are going to do. Below are some links to resources.
Tools
Quite often, QA teams use virtual machines to run tests. If you are going to use them, you'll need something like VMware or VirtualPC software. If you are going to use cloud computers, you'll need an Amazon EC2 account.
To control virtual and cloud machines, I'd suggest using Automated Build Studio. This tool lets you easily automate daily operations like running or stopping a virtual machine, launching or terminating cloud instances, building installation packages, copying files via FTP, restoring the databases and so on. It's a perfect tool for setting up the test environment and running the tests. At SmartBear we use this tool to run everyday builds and tests, and quite happy with it.
If you need programs for test planning and managing the QA and development processes, try using ALMComplete or QAComplete.
Also, you can try using AQtime if you want to check your application for memory or resource leaks, or get the source code coverage.
Working With TestComplete
To help you get started with TestComplete easier, I recommend that you visit our free TestComplete webinars:
http://smartbear.com/news/webinars-on-demand
and see the "Getting Started" section of TestComplete's Help system (if you have not done this yet) -- http://support.smartbear.com/viewarticle/12674/
A printable version of this section (Getting Started With TestComplete 8.doc) is located in the "<TestComplete 8>\Help" folder.
In addition to that, you can see our screencasts demonstrating how to use TestComplete features:
http://support.smartbear.com/screencasts/testcomplete/
We offer an online TestComplete training course:
http://support.smartbear.com/training/
You can also try training courses by other companies, for instance --
http://falafel.com/testcomplete/smartbearservices.aspx
You can find answers to your questions in the TestComplete FAQs, which are available on our web site:
http://support.smartbear.com/faq/
Some samples demonstrating how to use TestComplete features can be found in the following folders:
"<Users>\Public\Documents\TestComplete 8 Samples\" (for Windows 7, Windows Vista and Windows Server 2008)
"<Documents and Settings>\All Users\Documents\TestComplete 8 Samples\" (for other operating systems)
Here is a couple of books:
TestComplete Made Easy (scripting)
TestComplete Made Easier: Keyword Testing
Test Design and Frameworks
To get information on test design, you can read articles on our web site, or search the blogs. For instance, here is an interesting post:
http://blog.smartbear.com/post/11-07-11/automation-framework---a-new-table-driven-technique-using-testcomplete/.
Also, you can take advantage of the Mentor offer by Falafel:
https://falafel.com/store/mentor/smartbearmentor.aspx
