How do different training paths to multi-engine rating compare?
At the end of the day the choice might be made for you, as many flight schools will not let you fly their multi-engine aircraft until you have a private licence. 

People almost always begin on single-engine aircraft not just for cost, but also for efficient training. Multi-engine aircraft are complex beasts that require a lot of theoretical knowledge and practical skills to fly safely. If you're trying to get a grip on all that at the same time as just learning the basics of flying, you'll no doubt struggle and have to repeat lessons. Your first solo will definitely come considerably later, and many would agree that you learn the most about flying when you're up there by yourself. 

Granted, the more multi-engine time you can get, the better your future job prospects (generally speaking). Just remember that doing your early training like that will increase the time it takes to get your ticket. For a qualified pilot, a multi-engine rating only takes 10-15 flight hours of training anyway, so to me there isn't a huge advantage in getting one before your private licence.