Yes, before doing a formal search for registered and pending marks in the USPTO's database at www.uspto.gov, you can conduct a search for unregistered marks by using an Internet search engine. For instance, by entering your proposed name in the search field on Google (www.google.com), you will get a report of every instance where the name appears on Web pages that the Google search engine has indexed. However, for some marks -- for example, "Apple" -- a search engine may unearth hundreds or thousands of results that are of no value for your trademark searching purposes. For that reason you need to focus your search, using some of the searching tips provided at the Google site (or in Nolo book mentioned below). Alternatively you may choose to use a fee-based trademark search engine such as Saegis on Serion (www.saegis.com).
You can also search domain names being used by Web-based businesses at any domain name registrar. You can find a list of domain name registrars at ICANN.org, the organization that administers registrations (www.icann.org/registrars/accredited-list.html).
This type of informal search can alert you to unregistered trademarks that are being used to sell a product or service similar to yours. It won't help you too much, however, in determining which unregistered trademarks you are prevented from using. For help doing this, see Trademark: Legal Care for Your Business & Product Name, by Stephen Elias (Nolo).