Will a plastic feel less heavy when I put it in a bucket of water and carry it?
Fluids, like all objects, obey Newton's Third Law. This means that any upward buoyant force exerted on the plastic by the fluid has a counterpart downward force exerted on the fluid by the plastic. The force you must exert on the bucket & the fluid to keep it from falling is therefore increased by the amount of the buoyant force exerted on the plastic. If the plastic is in equilibrium, this is just equal to the weight of the plastic. Thus, the total weight doesn't change when you put the plastic in the water.
Your idea in the comment is basically correct. The situation is not fundamentally different from putting an object on top of another object that's sitting on a scale. When you do this, the reading of the scale will increase by the weight of the top object. The explanation is essentially the same as given above; just replace "buoyant force" with "normal force".
Assuming the densities of the water in the bucket and the density of the plastic are constant, and that no water spills out of the bucket when the plastic is placed in the bucket, then
$$W_{Tot}=ρ_{w}V_{w}+ρ_{p}V_{p}$$
Where $W_{Tot}$ = Total weight of the combination of the water and plastic in the bucket (ignoring the weight of the empty bucket, i.e., the container), in terms of the volumes and densities of the water and plastic, denoted by subscripts $w$ and $p$ respectively.
The first term on the right side is the weight of the plastic outside the bucket and the second term is the weight of the bucket of water without the plastic floating in it. Although the weight of the plastic decreases when placed in bucket due to the buoyant force acting upward, at the same time the weight of the bucket that now includes the plastic increases by an equal amount, so that the combined weight is still the same as the sum of the weights of each by themselves.
The above assumes that no water spills out of the bucket when the plastic is placed in the water. If before putting the plastic in the bucket the bucket was filled to the rim with water, then the water displaced by the plastic will spill out of the bucket, reducing the total weight. In this case, the reduction in total weight exactly equals the reduction in the weight of the plastic floating in the water.
Hope this helps.
If I'm carrying a bucket of water in one hand and a piece of plastic in the other, and then I decide to keep the plastic in the bucket of water (it floats). Will it feel less heavy in the second case?
There are at least two ways to consider this question.
First of all, does the bucket feel any greater force due to the object placed in it. The answer is yes. If you have 1 kg water bucket and add a 10 g bottle, then on a scale the bucket will now weigh 1010 g. Yes, the weight of the bottle is balanced by the force from the water. But that (extra) force is transmitted down to the bucket and to the scale (or your hand).
It does not matter if the object floats or sinks in the water. As long as we are ignoring the buoyancy of the air, then the bucket is going to weigh exactly the same as the sum of the bucket and the bottle.
The other thing to ask is if you could feel the difference as you walk. Probably not. Our senses act in a logarithmic fashion rather than a linear one. The hand can easily tell the difference between 0 g and 10 g, but it cannot easily tell the difference between 1000 g and 1010 g. The absolute difference is the same, but the relative difference is tiny. Your senses may tell you that the bottle doesn't weigh anything in the bucket, but that's just because you're using an inaccurate scale.