Even though it's not explained, there are a few other parts in the rulebook that might have helped you.
Generally, all bonuses on a card that are not listed below a horizontal line are "set up" once you play a card. +Card, +Coin, +Buy and +Action are the most common cases (with +Card being very obvious), but in fact it goes for all other effects, too.
This also goes for Treasures, by the way. If you play a Copper twice using
Counterfeit, the +coin the Copper produced are still in the pool, even though the Copper leaves play. This should actually be an example pointed out in the rules (i hope). Another Treasure that always leaves play immediately is
Spoils, also from Dark Ages.
It also works for more complex effects that will be visible only later in your turn, like
Coppersmith, which changes the amount of coin Copper produces this turn.
Using
Procession to play a Coppersmith twice will mean that the Coppersmith-effect, which affects your entire turn from now on, is "set up" twice. Even though Coppersmith leaves play because of Procession, when you play Coppers later that turn, each will produce 3 Coins instead of 1.
Be aware that some card effects might still not work as intended, though. Most of these will be because a card references being trashed or in play. If you allready trashed a card, you can't trash it again. If you allready trashed a card, it's not in play. An example would be using Procession on
Hermit. As the effect of Hermit that allows you to trash it for a
Madman is below a line ("when you discard this from play"), it doesn't get set up when you play it with Procession. And as Procession trashes Hermit before you can discard it from play, you can not use that effect to gain a Madman.
There are some more complex interactions, but as a rule of thumb: If the effect isn't under a line, it always happens. If it is under one, it only happens if some condition is met.
Edit: Unless you have the german edition of the Seaside expansion, where Duration cards have lines for no reason whatsoever.