To be honest, even as a local I get lost on those streets behind the casinos! Especially from Stratosphere (please don't walk around that neighborhood, in particular, behind the casino!)

From the Stratosphere, the easiest way would be to turn left on Sahara for just about half a block, and then turn right onto Paradise. Construction of the monorail has finished and you should be able to zip down to Flamingo (turn right) and get to Caesars, or keep going on Paradise and turn right on Tropicana to get to the Strip, then turn left to go to Mandalay Bay.

Then again, it depends on the day of the week and time of day. If it is not a weekend, and it is during the day, it will probably be just as easy to drive down the Strip and forget about shortcuts. Caesars isn't all that far, and even the long drive to Mandalay Bay isn't horrible - they have the stop lights set and sometimes you can zip all the way down the Strip with only a stop or two.

Plus, now with so many overpasses for pedestrians, the drive down the Strip is nowhere near as bad as it used to be. However, all bets are off on Friday and Saturday, so you might as well just plan to leave your hotel early to get where you are going on time.