The rules for lifting and carrying are on p.176 of the PHB. In a nutshell you can lift 15x your strength score and drag 30x.
If your total load is less than your lifting capacity you can pick it up and carry it around. If is greater than that up to your dragging capacity you can push or drag it or lift it over your head but not move with it. If it is heavier than that you can still drag it or carry it but your speed drops to 5 feet.
As an example, a fighter with 18 strength can lift 270 lb. and drag 540 lb. and is probably carrying about 100 lb. of his own gear. This means he can lift a creature and their gear if the combined weight is less than 170lb. and drag them if less than 440 lb. The pathetic wizard with 8 strength with 20 lb. of his own gear can lift 100 lb. or drag 220lb.
PC races weigh between 37 lb. (Halfling/Gnome) and 380 lb. (half-Orc) plus the weight of their gear; anywhere from 20-120 lb. so this is a relevant consideration.
If the thing you want to move is an object (which includes dead but not undead creatures) then that's all you need to worry about.
If the thing you want to move is a creature than things get more complicated.
If the creature is not resisting your attempts to move it around then you are not grappling. For example, carrying an ally out of the combat or if the creature is not in a position to resist; see below.
If you are grappling, then in addition to the weight restriction, your speed is cut in half. If your speed is now less than 5 feet and you use the grid variant, because of how speed is rounded, you can't move and so can't drag or carry your opponent.
As to the specific conditions, by the RAW:
- Prone creatures can still resist you, you would need to grapple them
- Restrained creatures can still resist you, you would need to grapple them; even though the princess is bound she can still wriggle like the dickens and stick her elbow in your eye if you're not careful.
- Incapacitated creatures can still move so would need to be grappled.
- Stunned creatures, even though they automatically fail Strength and Dexterity saving throws can still make Strength and Dexterity checks, indeed they have no impediment on them so are just as hard to grapple as if they didn't have the condition, and so can resist your grapple; you need to grapple them. Remember also, that a grapple check is an ability check as well as a "special melee attack" so the grappler gets advantage on their roll.
- Paralyzed ditto. However, you need to decide if someone in this condition can actually resist a grapple.
- Unconscious ditto.
You can decide that some of these RAW as stupid, as indeed they are, and house rule something else.