Things I check when troubleshooting a network issue with a provider. Note here that 'ping' means 'do an extended ping, sweeping a range of sizes, and sending at least 1000 packets at each size, looking for latency, loss and errors'.
- Ping your ISP's hand-off from a device on the same subnet. (Confirms that there are no L2 problems between you and your ISP).
- Ping your ISP's hand-off from a device on one of your internal subnets. (Confirms that you have no routing/security issues between your internal network and your ISP)
- Ping an IP in your provider's network. If you can't find one that's publicly listed, or the helpdesk won't give you one, do a traceroute to something in the internet, and use the second or third hop. This confirms that routing between you and your ISP is configured correctly.
- Ping an internet IP. I normally use 22.214.171.124, one of Level3's DNS servers that is easily remembered. This confirms that routing from your ISP to the specific IP you've pinged is correctly configured.
- Ping an internet domain name. This verifies your/your provider's DNS configuration is correct.
If steps 1 or 2 fail, it's most likely a problem on your internal network.
If step 3 fails, there would appear to be a problem in your provider's network. Note, though, that if you get no response at all then your ISP may have simply blocked ICMP to/from their infrastructure addresses.
Step 4 failing indicates a potential problem between your ISP and one of their upstream providers.
Step 5 failing indicates issues with name resolution.
Hope that helps.