You must draw right away.
Page 6 of the Rulebook:
Common terms used on the Action cards:
"+X Card(s)" – the player immediately draws X number of cards from his
Deck. If there are not enough cards in his Deck, he draws as many as he
can, shuffles the Discard pile to form a new Deck, and then draws the
rest. If he still does not have enough cards left after forming a new Deck,
he just draws as many as he can.
You must immediately draw the cards. You cannot wait to draw the cards after you have performed the other phases of your turn, or perform other actions on the card after the "+X Cards" line before drawing the cards.
Also note your specific example is clarified in the rules on page 7 (emphasis mine):
Spy – Spy causes all players, including the one who played it, to
reveal the top card of their Deck. Note that you draw your card
for playing Spy before any cards are revealed. Anyone who does
not have any cards left in their Deck shuffles in order to have
something to reveal. Anyone who still has no cards to reveal
doesn't reveal one. If players care about the order in which
things happen for this, you do yourself first, then each other
player in turn order. Revealed cards that aren't discarded are
returned to the top of their players' Decks.