Join Date: Oct 2004
Location: DC / Maryland suburbs
Bikes: Homebuilt tourer/commuter, modified-beyond-recognition 1990 Trek 1100, reasonably stock 2002-ish Gary Fisher Hoo Koo E Koo
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First of all, welcome!
Secondly, the problem is probably that the shift lever is too loose. The rear derailer tries to pull back to the highest gear, and the only thing that stops this is friction in the shift lever. Look on the side of the stem shifter unit: there should be a large screw on either side. Tighten up the screw on the side that is loose, until there is enough friction in the lever to prevent it from springing back. And if that doesn't work, I have a few spare stem shifting units... yours for the price of postage.
As for the rear tire: I suggest replacing the inner tube, as it is probably cracked or punctured. A new tube should only cost $2-5. Search the forums for advice on how to change it, or visit the "Repair Help" section of www.parktool.com
I hope you enjoy your bike. The old Raleighs are nothing high-tech, but I do love them. And if you ever decide to upgrade, you'll have plenty of people here willing to give you advice