I would like to start by talking about "BGP Looking Glass" first. "BGP Looking Glass" is basically a category of Internet Gateways (Cisco Routers, Unix/linux based platforms, Juniper Routers etc) running BGP. The idea is simple, many service providers hosts these internet gateways and gives limited access to CLI/GUI of those to the customers and sometimes even to anyone who needs it's access. This is a great tool when some network engineer wants to see if the routes advertised in BGP at customer site are reaching to internet and also if prefixes looks to Internet world same way as expected. It's also good tool to practice the wonderful world of regular expressions. I mean as you have access to a ISP router running full blown BGP config so what can be better than that to test your BGP regular expression skills and to see how other "show bgp" commands gives you tons of output on live BGP router.
Here is the address of one BGP Looking Glass Server which I use most of the time these days "telnet route-views.routeviews.org"
There are many more, even they can be categorized in terms of Service Provider who hosts this. Like if you want to see how your routes looks like when they reach inside network of SPRINT. In such case you can search through your favorite search engine for the BGP looking glass server hosted by SPRINT.
Also there is a great document on BGP Troubleshooting by friend Ruhnn which can be found at :
For more understanding of BGP from vendor independent point of view, you can visit : http://www.bgp4.as/
Hope this will help, more on troubleshooting will arrive soon :-)