2004 Honda Accord Alternator

My accord started to turn off while driving, it take a couple of seconds off and then it will turn right back on. When I took it to the dealership they said it was the battery I purchased and installed a new battery on the car, the week after the car kept turning off while driving, the radio would also turn on and off by itself and the lights will dim before the car would turn off.
they advied me that it was probably the alternator, the car has not been used for about a month and now that I went to turn the car back on it started fine but the engine rev's up and down by itself.

any suggestions reagarding this issue will be appreciated. Thank you.