I have trying to get a better mental big-picture of how the internet works I stumbled across the Wikipedia article about Tier 1 networks that really helped build my picture. This lead to Wikipedia'ing about interconnect agreements and especially internet exchange points which helped to greatly clarify the mechanisms of the internet. I found www.internetexchangemap.com which helped with a physical picture.
Exchange points are not the be-all and end all of peering. They are a mechanism to allow the costs of a physical connection to be shared among multiple peering relationships, the downside being that you have to pay for the exchange points switching infrastructure and organisational overhead. So low traffic peering relationships tend to go via exchange points while high traffic ones tend to go via dedicated private links.
When a provider buys "transit" (that is the service of providing connectivty to anywhere on the internet) then again that connection is almost certain to go over a private link.
There seems to be a big difference in attitude here between the US and Europe. The big access providers in the US tend to be very monopolistic and have the attitude that peering with "little guys" is giving away for free something that they could be charging for. In many cases those big access providers are also teir 1 transit providers.
On the other hand in Europe there is more competition among ISPs and the individual access providers are generally smaller and have to buy transit to reach large parts of the internet. The result is that there is more incentive for the access providers to peer directly with the content providers and internet exchanges flourish.
how does an AT&T end user request make it to a web server on a Verizon network?
Given that AT&T and Verizon are both teir 1 ISPs it will almost certainly pass over a private peering link between them. That link may well not be especially local though (you can often get hints as to the geographic route from the hostnames in traceroute).
A related question: I'm also wondering what the Milwaukee Internet Exchange means in their FAQ when they discuss how you can go there and "connect" (www.mkeix.net/faq/).
They don't explicitly list minimum requirements on that page but in practice they would almost certainly expect you to have your own IP space and AS number before allowing you to connect. I expect they would also at the very least consider it strange if you applied to connect to the exchange without first acquiring transit connectivity to the Internet in general
Note that connecting to an exchange doesn't necessarily let you interconnect with everyone on that exchange. The actual peering relationships are still at the discretion of the individual member.
The lack of requirements and fees listed on that page suggests they are a fairly minor internet exchange. Most of the big ones will have a lot more formalities and reccuring fees.