Minecraft – Leaving minecarts without moving them

In every attempt I've made at using powered carts, they've always come up short in terms of ability to push loads, not stick on things or travel far distances.

It's possible that changing the pushing device from a powered cart to nearby booster track would solve the sticking problem. There are many, many booster configurations that might fit the particular problem you're trying to solve.

The basic principle of starting minecarts with Powered Rails is that the minecart needs to "know which way to go". By this, I mean there must only be one possible way for the minecart to move.

A sloped Powered Rail works because the minecart knows to move downhill. Another way to do this without sloping the rail is to put a solid block on one side of the Powered Rail. By doing this, the minecart should start moving the other way when the rail is powered.