You can attribute properties to things for e.g. colour, smell, a name etc. One such property is the ratio of applied net force $\vec F$ to induced acceleration $\vec a$. Like other properties, you don't expect the ratio to be independent of almost everything: it could depend on the material, the place of doing the experiment, surroundings, the temperature who knows? What does one even mean by the ratio of two vectors - it mayn't even be a scalar.
Turns out that the ratio is remarkably independent of the object's other properties its calculated for. It doesn't depend on the type of element the object is made up of*, nor its temperature** nor where the object is. Moreover, there are frames of reference in which its completely characterised by a corresponding single scalar number for every object.
This property is called mass.
Human beings have developed a sense of measuring mass in the form of inertia. You push and see if something moves. The harder the push, more the inertia. The ratio discussed above measures precisely that: amount of push per unit movement. So inertia is commensurate with mass.
Point to note here is that one has to push to get a sense of inertia. People can push stuff over all kinds of things: push a cart in a meadow, push a car on a highway, push yourself on a skating rink, push yourself on dry sandpaper. Turns out there is a different inertia to things depending on the surface. So is the mass different?
"No, No", says the experimentalist. We blame the variation on extraneous circumstances and label that as friction. Its the surface to blame not the object.
You see, the sense of inertia that one has is not as controlled of a scientific property as the mass. If one measures the ratio, far, far away from anything and everything (don't ask how), one would find that its just one scalar $m$.
Its therefore easy to think, as you say, that obviously things would be harder to move around on a planet with stronger gravity. That is what intuition would say, developed on a planet with just one gravity.
But you would be wrong. You see, its again not the mass which is making things difficult here. Your sense of inertia is off because in the mental picture that you have, gravity is acting invisibly to make life harder for you.
Saying moving a block around is harder in stronger gravity and so its got more inertia is like saying a car stuck in dried concrete is heavier. Yeah its harder to push, but it is still the same heavy - the same mass. Its just being held in place very strongly. You don't say that it's heavier or its got more inertia: you just say its being held down firmly.
After pushing a block around in a stronger gravitational field, you'd probably be (very) tired. So you lay down the block and go to sleep. And then you try to lift it up in the morning.
Human beings have developed a sense of inertia when things that have been laid down are picked back up. We call it weight. Since lifting is just like pulling(pushing)-just in some other direction, weight feels like inertia to us. If something weighs more it most certainly has more inertia and therefore more mass, so we feel.
And that is the root of all confusion.
You see, unlike mass which has the remarkable property that its independent of the amount^ of the applied force, weight doesn't.
In fact one can make weight zero. While lifting such things one wouldn't have to apply any force at all. To push(accelerate) them though, one would. So the inertia would exist even without the weight. Associating inertia with mass therefore makes more sense than weight.
Alas most of the earthlings are earth bound, where they can't really change their weight without varying their mass, so intuitively they would always be the same to us and associating inertia with either would do no harm. Astronauts would beg to differ.
In short , things are harder to in stronger gravity because they weigh more not because the have more inertia.
*In the sense that you can have the same mass to be made up from any elements. Changing the elements will, of course change mass.
^Even more remarkably it doesn't depend on the nature of the applied force.