When moving web hosts you are going to have some downtime unless you are using the name-servers of your registrar, in that case you just need to change the ip address of your new host/VPS server. Below is a list of my favorite DNS tools:
Open DNS Cache check
This tool from OpenDNS helps you check what server OpenDNS customers see when they request a domain. It includes their servers across the world including North America, Europe and Asia.
Pingdom DNS tool
I highly recommend the tools from Pingdom including their DNS server checker. Just enter your domain to check if its DNS servers are configured correctly. After the test it gives you a Permalink to allow you to share the test link. The results are either in Basic view mode for easy reading or Advanced mode for you techies out there.