I live in a house that was built in the 1930's. I'm paranoid that the old pipes for hot and cold water(for the kitchen, shower, etc..) are contaminating the water. Is this something I should be worried about? What is the best way to test the water that flows through the pipes in my house?
Water testing (from what I've seen) is cheap and easy enough that it's worthwhile to do if you are worried at all. If you're in the US, check with your local Department of Natural Resources to find out where to get a sample kit. Where I live, it's $10 for a container and testing from my State Health Lab.
Typically, you'll need to sterilize a faucet before you take a sample (either using a torch or flushing it with bleach) and you'll need to drop it off for testing fairly quickly (24-48 hours). This is third-hand from my home inspector (and I'm sure different labs have their own requirements for prepping and collecting the sample) so definitely read the directions you get with your kit. I recently purchased an older home with a well and I'll be doing a test myself this week so I'll be able to follow up with actual experience soon :)
Your locality should test water in the area on a regular basis, but this is via sampling, so problems specific to your home will be missed.
You can almost certainly send a sample off to a lab and have it tested, and the EPA may point you towards someone that will perform various tests, particularly for lead. But since we're all about DIY, the easy solution is to pickup your own test kit which will help identify problems in your own plumbing. You can find these on amazon (search on "drinking water test kit" if the link stops working), and if you're lucky, the local HI store may even stock these.
Note: I've never used one of these kits before, so I can't personally vouch for their reliability. However, I have sent samples of water to my municipality in the past (I didn't receive an individual report back from this). For best results, you need to sample the first water that comes out of your tap after sitting overnight.
In some areas, you can even get a free water test. Here in Toronto, we got a free water test and our lead levels were higher than I'd like, although still within the range of acceptable.
Seems like finding out how much it would cost you to get a water test, and then getting one done if it's not too expensive is a reasonable step to take, particularly if it'll give you some peace of mind.