I do best in physics when I can make sense of the units that accompany values, and I do this by visualizing in my mind what is happening. Take for instance, $v=\frac{s}{t}$. When I think of velocity I can visually see an object moving a distance, $s$, over a certain time, $t$. What I struggle with though are units that aren't 'x per y'. Like meters per second.
Take impulse: $N\cdot s$ or Newtons: $\frac{kg \cdot m}{s^2}$. How can I visualize a Newton-second or a Kilogram-meter.
If you look at a product or division, the way is to start with one thing.
Look at 'Newton-second'. This means a newton applied for a second. A newton is a force, a shove, so to speak. You shove something for a second, and it starts moving. If it weighs a kilogram, the effect of a newton-second is to make it move at 1 m/s. If it weighs 10 kg, it will move at 1 dm/s.
A metre-newton of torque, is where you pull on a handle with newtons of force, at a distance of metres. The old rusty nut isn't going to turn too fast unless you get a longer spanner. Then you need to pull more to get the same angle, but it's a bigger force.
If you get something like 'volts per metre', it's a gradient. 'per metre' often gives gradients, or weights of wire. So volts per metre here is a gradient. The unit translates to newtons per coulomb, and it's the electric force doing the shoving. Here the size of the handle is in coulombs, and the newtons come as force.
Sometimes it's better to look at how the quantity is defined, and then see if you can make the units to work. One quantity I came across was 's/m' of permeability. It works like this. The actual thing is 'flow of liquid per area, divided by the driving pressure', so (kg / m² s) / Pa. But when you put Pa = kg / m s², the division gives s/m.
Another unit is 'W/m K'. This is thermal conductivity, or energy flux per driving temperature gradient. What it means is (energy / area / time ) ÷ (temperature / thickness). The fps unit is Btu / ft² h ÷ °F / in, or the unit Btu in / ft² hr °F. Replacing these with their metrics, gives J⋅m/m² s K, or W/m K.
Physical units essentially keep track of the measurement processes needed to determine the quantities you are dealing with. Once you understand the procedure to derive those quantities, the actual units are irrelevant, especially because changing the measurement apparati may result in different definitions of the units themselves. The standard example is that of special relativity: if you fix the velocity of the signals you are dealing with (say, $c$) then measurements of times can be addressed to measurement of distances, and viceversa; therefore the two quantities may end up being defined with the same units.
This said, the attitude of trying to visualise things is unfortunate: not everything can be visualised (but can be understood, on the other hand). I do not see the difference, in your example, between visualising the velocity and the impulse: they are equally hard to imagine unless you know how to perform an underlying experiment to determine them. But if you did have an experiment to calculate the impulse then it would be straightforward to understand what it is.
I find that newtons are easiest to intuitively understand if I think of it as corresponding to how much force I have to put on an object to keep it from falling. The heavier the object, the more force I need, but in a different gravitational environment, I would not have to use as much force, even though the mass of the object would not have changed.
