So the verb pair for "to wash" is "мыться/умыться"?

мыться is to wash your whole body, умыться is to wash your face

if I write "сходил покупками" instead of "сходил за покупками" is it wrong? And why "сходил" and not just "ходил"?

сходил покупками is wrong. Покупками is a noun so it can't describe the verb directly. So you need to add something between them such as a preposition in this case
сходил за покупками - literally - went for some commodities.
сходил за покупками or ходил за покупками doesn't matter in your case. Both would fit in nicely

what is wrong about "Я буду быстро возвращаться."? why you wrote "Я скоро вернусь". My version is unintelligible?
Well, everything is wrong with your original phrase here. It doesn't work for this particular situation. First of all, you need a perfective verb there because it's a one time action, you will return there only once, and also the action is not in the process. As for быстро/скоро. I guess you could have said "Я быстро вернусь" but it just doesn't sound good at all.

@Heterocera: If I am not mistaken "отпуск" is an organized tour. However, I do not know if "Путешествие"/"тур" means something so different.

Отпуск is that special prolonged time (usually a month) you get once a year to rest from all the tiredness you job brings along and still get payed through that time This thing is a law.