There are several tools you can use to do a name server lookup. Applications like CyberKit can be downloaded at software distribution sites. If you decide to use one of the DNS lookup tools, you should start out by querying one of the root servers. Once you query the root server, just follow the name servers until you get to the name server that hosts records for your domain.
Some versions of Windows come with the NSLOOKUP feature built it. If you have this tool, you will be able to use it via the command prompt. Read the Knowledge Base article on How Do I Use The NSLOOKUP Tool Provided With Windows? for basic instructions on using this program and the root servers information.
Most of the available DNS lookup tools require some technical knowledge. Therefore, you may want to use a more simple means of checking your DNS records. There are online tools out there which will do all of the work for you: they will check the root servers, follow them to the name servers, and return the information. Some of these sites are:
Just type in your domain name (without the www.) in the box provided, select the type of information you want, and submit the query.
If you check the DNS information for your ISP's name server, you will know whether or not the new DNS information has propagated to that server. Please note that this does not mean that propagation is complete, as there may be other DNS servers out there which don't have the new information. Read the Knowledge Base article on What Is DNS propagation? for more information.