The expressions "to wear clothes" or "to dress up" can be best translated with the verb llevar:

Marcos lleva una camisa – Marcos is wearing a shirt

This is the most frequent and the easiest way to describe clothes. However, there are two other constructions:

Marcos está vestido con una camisa – Marcos is dressed in a shirt
Marcos viste una camisa – Marcos is dressed in a shirt

The verb vestir usually means to "dress the children":

Voy a bañar a los niños, y luego los voy a vestir – I’m going to bathe the children, and then dress them

It can also mean "to dress in a certain way", for a general description of style or color range of clothes:

Marcos siempre viste de negro – Marcos always wears black
Marcos viste muy elegante – Marcos wears elegant clothes

Pay attention to prepositions! Compare:

Vestirse con una camisa – To be dressed in a shirt
Vestirse de negro – To wear __ black

The verb ponerse is translated as "to put on":

Para salir, Marcos se pone un abrigo – To go out, Marcos puts on a coat

When used without a reflexive pronoun, the verb poner means "to put some clothes on someone":

Voy a poner al niño este abrigo – I’m going to put this coat on the child

The reflexive verb vestirse means the process of dressing and is translated as "get dressed" (without a direct object):

Marcos se viste de prisa y sale de casa – Marcos gets dressed quickly and leaves the house

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