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:
Can a Warlock benefit from holding two rods?
Yes, a warlock can gain the properties from two rods but he still can only use one to make an attack.
Likewise, from the rules on Implements (PHB2 p220, PHB3 p219):
The implement keyword identifies a power that can be used through an implement, and the implement must be a type wielded by the power’s class. (...) If you wield a magic implement, you can add its enhancement bonus to the attack rolls and the damage rolls of implement powers you use through it.
And the excerpt of the rules on Holy Symbols (RC p274):
Associated with the divine power source, a holy symbol represents a deity and bears an emblem or an image of the god. Unlike other implements, a holy symbol can be used whether it is held or worn. If a creature wears or holds more than one holy symbol, none of these symbols function until only one remains (this rule is usually relevant only to magic holy symbols).
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.
Dual Implement Spellcaster
Prerequisite: Dex 13, any arcane class
Benefit: When you use an arcane attack power and you are wielding a magic implement in each hand, you can add the off-hand implement’s enhancement bonus to damage rolls. Both of your implements must be usable with this power, and you must be capable of wielding both implements, to gain this benefit.