schleifen
to grind / sharpen
To smooth a surface through friction or sharpen a blade.
Ich muss das Küchenmesser schleifen, es ist ganz stumpf.
I need to sharpen the kitchen knife; it's completely dull.

to grind / drag
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to grind / sharpen
To smooth a surface through friction or sharpen a blade.
Ich muss das Küchenmesser schleifen, es ist ganz stumpf.
I need to sharpen the kitchen knife; it's completely dull.

This verb has two conjugation patterns. In the meaning 'glätten, schärfen' (to smooth, to sharpen), it is strong: schleifen, schliff, hat geschliffen. In the meanings 'ziehen' (to pull/drag), 'drillen' (to drill), and 'abreißen' (to tear down), it is weak: schleifen, schleifte, hat geschleift.
Der Handwerker hat das Holz glatt geschliffen.
The craftsman sanded the wood smooth.
When the object is a blade or a surface that is being worked on, 'schleifen' (strongly conjugated) means 'schärfen' (to sharpen) or 'glätten' (to smooth). Typical objects are knives, scissors, parquet flooring, or gemstones.
Er schliff die Klinge seines Taschenmessers an einem Stein.
He sharpened the blade of his pocketknife on a stone.
When a heavy object is moved laboriously across a surface, 'schleifen' (weakly conjugated) means 'ziehen' (to pull/drag). A direction is often given.
Die Kinder schleiften den Schlitten den Hügel hinauf.
The children dragged the sled up the hill.
In the meaning 'drillen' (to drill), 'schleifen' (weakly conjugated) is used almost exclusively in military or sports contexts for hard training. It has a very negative connotation of excessive harshness.
Der Trainer schleifte die Mannschaft bis zur völligen Erschöpfung.
The coach drilled the team to the point of complete exhaustion.
The meaning 'abreißen' (to tear down) for buildings is rare today and mostly historical. You find it in texts about castles or fortresses that were destroyed as a punishment.
Der König befahl, die Burg des verräterischen Grafen zu schleifen.
The king ordered the treacherous count's castle to be razed.
to pull / move
'ziehen' and 'schleifen' can both mean moving something by applying force to it. 'ziehen' is the more general word: you can pull something toward yourself or pull it along behind you. 'schleifen' is more specific when something heavy or awkward is dragged across the ground and the action seems effortful. If only the direction or movement is important, 'ziehen' fits. If you want to emphasize that something is rubbing along the ground, 'schleifen' is a better fit.

Nach dem Umzug schleifte Paul die alte Matratze in den Keller.
After the move, Paul dragged the old mattress into the basement.
Max zieht den Stuhl näher an den Tisch.
Max pulls the chair closer to the table.