In order to clear a sink drain clog in an upstairs bathroom this weekend, I needed to remove the shutoff valves to get clearance to remove the P trap. This entailed turning off valves in the basement for the lines that supplied the upstairs bathroom. Everything went back together again and, after some initial spluttering to rid the air in the lines when I first turned the water back on again, all seemed well. However, sometimes if I open the faucets only part way, a hammering starts in the pipes and the water turns off/on about 4 times per second. If I open the faucets further, the hammering stops. This happens in the sink faucets and the shower, which are fed by the same supply lines and happens in both the hot and cold sides. This never happened before I shut off the water.

Is there anything that I can do to get rid of this problem without having to install water hammer arrestors? Since this never happened before, I don't want to tear into my walls if I don't have to.

That hammering is due to air included somewhere in the circuit.
Open all faucets to get the air out.

This is neither a "water hammer" or a "air in pipes" situation and you shouldn't have to tear into walls to fix it. The sound you hear is caused by vibration. A piece of washer or mineral build up, that you disturbed when you took the stops off and replaced them, vibrating against the seat as the water rushes by. Start the vibration again and try to track it back to the valve that it's the loudest in. When you locate the valve shut the water off and open it up. Place a pan over it to prevent splash and turn the water back on for a minute to flush out the supply lines. This may take a little detective work to track down but my money's on one of your bathroom valves. Good luck and let me know how you make out. Tom