First, you can properly interpret a kg*m/s^2 by thinking of it like this: one newton is the amount of "force" it takes to accelerate one kilogram one meter per second squared. Further, you can interpret momentum as this: one kg*m/s or one "momentum" is the amount of momentum one kilogram going one meter per second has. Momentum is just a qualitative idea that we thought would be useful to measure and so we assign it to things as something quantitative. The only reason we combine these "things", increasing/decreasing things by certain factors, is to provide a certain insight into the phenomena itself or insight into another phenomena.
In terms of interpreting these "units", don't forget the key principle behind them; multiplication and division, which are two forms of the same thing! Division is a form of multiplication rather, being the multiplication of the inverse. So, it is fair to say that you can think of these multiplied quantities just as you've been thinking of these "divided" quantities.
When you say you understand a "meter per second" m/s, you are saying you understand a "meter per one-over-a-second" or a "meter times one-over-a-second" m*(1/s). It just so happens, in terms of context, you understand it's principle. Now when we think of m/s^2 things seem to get a little trickier. But they don't. All this is saying is "a meter per second per second". Now you don't find it strange that we have those two 'seconds' multiplied together in the denominator, because we know what it means; we are "accelerating" one meter-per-second per every second that goes by.
All units can be thought of this way. It takes a bit of imagination, you just have to think about it! A meter squared is a meter for every meter, a meter-in-the-x-direction per every meter-in-the-y-direction. Per just means "for each".
Energy is an abstract quantity with a peculiar quality; it just so happens to be conserved in many different forms. Thus, it has many different units that all represent the same thing (this is where the physicists are guilty of being a little nutty in a sense that they can't get the units straightened out!). As a result, you may think of energy any way you want. Convert joules to calories if you'd like! Or even Celsius - if it really makes it simpler for you to understand. Or you can think about it in context; a joule is the amount of energy it takes to bring a newton of weight up one meter from an arbitrarily defined surface - which now represents the amount of energy that that newton-weighted object has in terms of "potential energy" - which is said to be "stored".
I hope this is of any use to you so many years after...