A good way to check for available domain names is through a domain name search engine, such as GoDaddy's Bulk Registration page, which is shown below.
It will let you enter a bunch of domain name possibilities (one per line), and when you click GO, it will tell you which domain names are currently available.
If you'd like to search across multiple TLDs (Top Level Domains), such as .COM and .ORG, just enter the domain name without the .COM or .ORG extension into the box on the left. You can then select as many TLDs as you'd like from the check boxes on the right.
In the example shown above, we're searching for my-first-choice, my-second-choice, and my-third-choice only from the .COM TLD, since we only selected the .COM check box on the right. The result of this domain name search is shown in the image below.
The domain names that are unavailable are shown at the top. In this case, my-first-choice.com was already taken. However, my-second-choice.com and my-third-choice.com were available, so they were added to My Selected Domain Names, which is shown in the middle.
If you want one of the domains that were available, you can register it by clicking the Add & Proceed to Checkout button.