Can I put a lower geared cassette on my bike? I bought a bike recently that has a 34 front 28 rear gear ratio. It is an Elfama E2500, which is a Korean brand I believe. It has a Shimano Claris (which is rated for up to 32 teeth) rear derailleur and an 8 speed 11-28 cassette in the back. 

I would like to go with as big of a gear as possible for climbing, somethine like 11-32 which is the biggest I can find in an 8 speed. Something like this... http://goo.gl/eLbluH

Will this fit? Is there some other variable I am missing? Would it be possible to go up to a 10 speed cassette without changing too much on the bike?
Changing to 10 speed gets real expensive real quick, because you need new chain, cassette, shifter at a minimum.

32 tooth is 4 more than 28, so its a ~14.3% increase.  That's quite a lot - you will need extra chain links, and your rear derailleur will need adjusting and may not work with the extra teeth. It's hard to be sure without trying it.

Your other possible change is to shrink the smallest of your front chainrings.  This may mean slower changes between rings, or the FD may not have enough capacity.  You'll also need a medium or long cage rear derailleur to help take up the difference in chain length.

Final option is to grin and bear it.   Climbs up hills aren't fun.  They're supposed to be a workout.