WireShark has a steepish learning curve, but with that tool you can identify where things go wrong when you surf to these problematic websites. This software sees all network traffic going through your network and you can gain a lot of deep insight in what your problem is and where it occurs.
Tools like traceroute and perhaps nslookup (regarding DNS) are the first troubleshooting tools you should use to find out about simple networking issues. But for a more thorough investigation, use: Networkminer or when things are getting complicated: Wireshark.
Both can be obtained for free, but require that you install the latest version of WinPCap (also freeware). The pro version of Networkminer is more powerful, but comes with a hefty price-tag: 900USD, but the free version is likely already helpful enough.