felvesz – to put on
Felveszem a kabátot. – I put on the coat.
Felveszik a famert. – They put the jeans on.

levesz – to take off
Leveszem a kabátot. – I take the coat off.
Leveszik a farmert. – They take off the jeans.

Talking about your shoes you can also say: felhúz
Felhúzom a cipőmet. – I put on my shoes.
Leveszem a cipőmet. – I take off my shoes.

NOTE!

Felvesz and levesz are phrasal verbs: fel + vesz AND le + vesz. As vesz is irregular in the past tense, you should pay attention to it: vesz > vett

Levettem a pólót. – I took off the shirt.
Felvettük a zoknit. – We put on the socks.