Συλλέγω belongs to a formal register.
The more usual expression for "I collect (stamps etc.)" is κάνω συλλογή (γραμματοσήμων κλπ.)
And yes, there is also μαζεύω, but without an article (μαζεύω γραμματόσημα etc.)
μαζεύω χρήματα και λουλούδια then
I think one can use the article with μαζεύω, but with another meaning. To pick up or gather, not to collect. I must say to collect is also to gather I think, with the article. the use of the article changes the meaning. No?
And then without context ' I collect everything' can also mean 'I gather everything' . But in Greek you say μαζεύω τα πάντα .The article is always there.
Quite right. Μαζεύω can perfectly well be used with the article, but then it refers to a definite set of things. Μάζεψε τα γραμματόσημα που είχαν πέσει κάτω = he picked up the stamps that had fallen on the ground. O μικρός μαζευει τα γραμματόσημα της εισερχόμενης αλληλογραφίας = The boy collects the stamps (found) on incoming mail.
Note also the expression τα μαζεύω, which means "to pack up one's stuff and leave".
To παν / Τα πάντα is a special case. It is always used with the article.
Also, μαζεύω χρήματα means "I am saving money" (to retire to the Canaries, e.g.) If you mean "I collect coins and banknotes", you must say "κάνω συλλογή νομισμάτων".