So far I haven't found a good all-digital solution. I typically use an audio splitter at the computer, and a whole house FM transmitter plus speakers so I hear the sound locally and through FM radios I place wherever I want the sound.
Since everything is transmitted real time, there's no sync delay as you move from room to room, and the transmitter I have is strong enough to receive using a boombox out in the yard while I'm working outside, without risking any expensive equipment in case it gets kicked around the yard or rained on a little bit.
I use a modified Belkin Tunecast transmitter. I disassembled it, and added a length of wire to the antenna connection to stretch the distance it transmits a little. There are some legal ramifications to this depending on the country you live in. If you want a more professional solution, there are a few places that sell whole-house FM transmitters fairly inexpensively.
You could also connect the transmitter to the airplay receiver, entertainment center, iPad, Android phone, windows PCs, or any other audio device, and the receivers are cheap - you can get an FM radio from the dollar store for a dollar, and amplified speakers for a dollar, so expanding the system is cheap. It's very flexible.
The biggest drawback is quality. It's not bad, but you can't expect even CD quality from it, nevermind high quality audio. It's ok for music and voice, but audiophiles will be offended. Airplay is CD quality, and the difference would be noticeable.
Best Answer
The short answer is, that you are doing the right thing, the Apple TV should just wake up when it gets an AirPlay stream coming at it.
My Apple TV (720p) one is having similar issues, you have to wake it up to stream to it, and it has other issues.. I would suggest a factory restore which is what I am going to do.
It will loose all your preferences, but should fix the issue. Worst case you could also look at reinstalling the firmware using a Micro USB cable and iTunes on your PC or Mac.
Instructions on Apple.com for both methods