I solved my problem, and
Problem:
Lazy web designer (me) wants to use bitcoins without dealing with installing bitcoin on a server, installing a shopping cart interface, or using ugly merchant services with callbacks.
Solution for sending bitcoins:
Use the MtGox API (https://mtgox.com/support/tradeAPI)
Solution for receiving bitcoins:
1) Input a list of bitcoin receiving addresses to your database
2) Give a bitcoin address to a potential customer
3) Have the customer tell you when they have sent the coins and have at least 1 confirmation (you can choose a number higher than 1 if you are worried about double-spending)
4) Check blockexplorer to see if they sent the right amount (i.e. http://blockexplorer.com/q/getreceivedbyaddress/19hMEAaRMbEhfSkeU4GT8mgSuyR4t4M6TH/1) - the /1 is the number of confirmations you require
5) Give them what they paid for
6) After a reasonable amount of time has passed, you can re-use the address for another customer
You could avoid having a list of addresses and reusing them if one of the wallet services someday lets you get a new address via API call, but this will work for now.
Bad idea:
Selling bars of gold this way (owner of blockexplorer.com could rip you off)
Good(?) idea:
Selling naked pictures of your grandma this way (owner of blockexplorer.com won't bother)
Shameless begging:
If this info is useful to you, please consider a donation: 19hMEAaRMbEhfSkeU4GT8mgSuyR4t4M6TH
Maybe I'm the only one this lazy, but just in case, I added a page to the wiki: https://en.bitcoin.it/wiki/Lazy_API
Thanks to theymos for making changes to blockexplorer.com to make it even easier to do this. Because of those changes to better support what I want to do, I paid Theymos the 20BTC bounty. You can support blockexplorer.com too by donating to 1Cvvr8AsCfbbVQ2xoWiFD1Gb2VRbGsEf28