The way I work out the range? I have a mental image of what general curves look like (trig functions, quadratic, cubic, logs etc....), then in my head I identify where on these vague shapes are the minimum and maximum heights.
For example in a quadratic function y = x^2 + x + 2, you know it looks like a "U" shape, so its range is going to be infinite in the upwards direction (because the top of the U never stops going up), but bounded in the negative direction at the very bottom of the "U" shape.
The next mental step is figure out how to find that point. Sometimes you just need to know some properties of each curve (e.g. exp(x) cannot go negative, sin(x) goes between -1 and +1), then work from there.
For a quadratic "U" shape it's got a single minimum point, so you find it: differentiate (2x + 1) set equal to zero (2x + 1 = 0) and solve (x = -1/2), then find the corresponding height value (y = (-1/2)^2 + (-1/2) + 2 = +1/4 - 1/2 + 2 = 7/4), and you know that all other points in the curve must be higher than that. So the range is y >= 7/4.
A sanity check always helps when you've got time. We said the curve reaches a minimum of 7/4 at -1/2. So let's check what it is either side of this. When x = 0, y = 2 which is > 7/4 so that's good. When x = -1, y = 2 again, which is also in the range.
Practice and some experience, that's all it takes.
I might also add one useful thing to know, since you asked about the range of a y = ax^4 + bx^3 type curve. The shape of the curve is kind of dominated by the highest order term. So this particular curve has the shape of a quartic curve, which in general is like a "W" shape. Lots of vague maths going on here but it's served me well through grad school. With a "W", it's almost the same as the "U" of a quadratic, except you've got two minima, and it's not immediately obvious which one is the minimum, so you've got to find both of them and take the lower of the two as your minimum range. As before, there's no upper range because the two side edges of the "W" continue upwards forever.
One more hint: if the curve is linear (x^1) or cubic (x^3) (or any odd highest term), it has an infinite ranges, because when x gets hugely positive x^3 gets hugely positive, and when x gets hugely negative, x^3 gets hugely negative. This doesn't happen with the even ones.