Here's my collection of favorite DNS tools and techniques to troubleshoot issues:
- Great DNS report generator. (Huge report to find all kinds of problems in your DNS; use the "DNSreport" box there.)
- How to check that your change of nameservers was entered at the TLD's nameservers. (Use second part of that answer.)
- DNS delegation checker. From root nameservers down to authoritative nameservers.
- Finding your primary nameserver. (Essentially, look at the SOA record.)
- Wordlwide DNS cache status checker. A tool to check what DNS servers around the globe have what value (i.e. which have expired caches, and which still have old cached values.)
- Various DNS tools. (See "Domain Inspector" for a tool to look up all records on all nameservers, starting from the authoritative ones.
- Wikipedia on Domain Name System. Some good background knowledge about DNS!