This is an interesting topic; I posted an answer to a somewhat related question several years ago.
The best source I know of for obtaining climate data is the Ingrid system, run by the Lamont-Doherty Earth Observatory. This site has almost every large atmospheric and oceanographic data set ever collected. It will take you some time to learn how to use it, but it's well worth the effort.
Some highlights of the Ingrid system:
As for lunar phase information, you can easily get that from a planetarium program, like Distant Suns (for the PC), Starry Night (for the Mac), or Xephem (for Unix). See SEDS's Planetarium Software page for more info.
But really, all you need to know is that the cycle of lunar phases repeats once every 29.52 days. If you know the lunar phase today, you can figure it out in the past using this number. On the other hand, the Moon's period of revolution (with respect to the fixed stars) is once every 27.32 days. I couldn't tell you which is more likely to influence atmospheric pressure.
As for building a barometer, the easiest way to build a barometer is to make a glass tube at least 32 inches tall, closed at the top, fill it with mercury, and sit the open end in an open dish of mercury. Unfortunately, this old-fashioned sort of barometer is dangerous, because mercury is very toxic. Building any other sort of barometer is probably too difficult -- calibration will be very tricky. But you might be able to buy one from a scientific supply house, like Edmund Scientific.
If you insist on building one, my suggestion is to search in the magazine Scientific American. They have an "Amateur Scientist" column which regularly describes how to construct scientific apparatus, and a barometer is one of the most useful instruments one could make. In fact, I found a reference to "Making a barometer that works with water in place of mercury", in the April 1987 issue, page 114. Check this page for an index of the Amateur Scientist column.
I also found this site, which describes one way to build a barometer.
Try the links in the MadSci Library for more information on Earth Sciences.