If I'm a Cleric with the Student of Artifice feat, and I have a staff and a holy symbol, will my powers with the implement keyword receive both enhancements?
You only receive an enhancement bonus the one time. Note that each item says it grants an "enhancement bonus"; same typed bonuses don't stack (in this case the type is "enhancement").
Note: you don't need a particular class' implement to use powers from that class. Any implement you are proficient with will do.
No, they won't. You can only use the enhancement bonus of one implement - pick which one at the time of using your power.
The default behaviour for wielding multiple implements is that you can gain the properties of each implement. When it comes to using a power, however, you can only use one of your implements to use that power. Choose one, and add its enhancement bonus, and any other effects that matter when you're using the power through that implement.
The Student of Artifice feat doesn't change this behaviour.
Holy Symbols don't change this either. They may be special in that you don't have to hold them to use them, but (in response to the asker's query in comments) you don't get to circumvent this behaviour and combine them with other implements. The only difference between using a Holy Symbol and using a regular implement is that whilst a Warlock would have to lift his rod in his hand to use an implement power with it, a Cleric could leave his Holy Symbol tied around his neck, or wherever.
From the FAQ for the Player's Handbook:
Likewise, from the rules on Implements (PHB2 p220, PHB3 p219):
And the excerpt of the rules on Holy Symbols (RC p274):
Can I change this so I can use two implements at once?
Yes. There is a feat called Dual Implement Spellcaster, which by existing reinforces the default behaviour of only using one implement at a time, and allows you to use two implements.