in all the bamboozaling on the 4Cs its worth remembering there isnt alot of difference between the grades, and yes the cut is the most importent - as if it isnt cut well, it doesnt sparkle, and why would you buy a diamond that doesnt sparkle - you might as well buy a Cubic Ziconia from QVC. 
in terms of colour grades - 99% of the population couldnt line up a range of diamonds in order of grade from D to J, and a good rule of thumb i work to is H or higher is fine - it appears just as clear as a D and honestly you couldnt tell a difference.
Diamonds are graded face down, against a white background in controlled light conditions - it is also a graders oppinion, and the same stone sent to the same lab may sometimes come back with a slightly different grading (same for clarity)
But in the real world you will never be in those conditions - and a well cut diamond will reflect alot of the colours around it.
For instance put a D diamond in a yellow gold bezel setting (ie a setting where the edge of the diamond is covered by the rim of the setting) and it will look slightly yellow - where as a K diamond in a platinum 4 prong tiffany setting (the classic style of ring) would look alot white in comparison to the D in gold)
there are other factors that can also effect the face up colour of the stone - mainly the cut of the stone - but thats getting a little deeper into things than perhaps is necessary to the OP (the basis is that on stones over 0.5ct if the pavilion angle, that is the angle of the lower half of the stone, is greater than 41° then the stone will show the body colour of the stone more than a stone which is cut at a slightly shallower angle)
But Colour does not have any bearing on how 'bright' as stone appears - that is how well cut a stone is, tho as a proviso this is for stones in the D-K range - M N O then the yellow colour is alot more noticable, and beyond that we are into fancy yellow - and on differnet scales is the fancy colours blue, red etc. and these stones are cut to enhance the colour of the stone, not hide it.
Clarity - a bit like colour grading - to the naked eye clarity doesnt make much difference, it is only the wallet it makes a difference to.
again my rule of thumb is to look at VS2 and above - while taking SI1 on a stone by stone basis.
the clarity is looking at imperfections within the stone - these can be lots of things - usually specs of other material within the stone. Remember a diamond is formed millions of years ago deep in the earth - the imperfections are unique to each stone, and are its birthmarks.
taking 2 different stones with the same clarity grade you could have one stone with a single black inclusion, and the other with lots of tiny clear inclusion (a cloud) and depending on where that big black inclusion is - it could be in a postition where it can be hiden by a prong of the ring and there for invisable when set in a ring.
when looking - if the diamond is certified, then the cert should have a plot of the inclusions visable in the stone. Hold the diamond close and see if you can find them in the stone - if you cant then, why pay more for a stone with a higher clarity grade?
personally i bought my wife a pair of diamond studs - they are VS clarity - i needed to use my 30x loupe to pick out the clear inclusions in the stones, nothing can be seen with the naked eye.
again clairty is assessed by a grader under magnification in controlled lighting which highlights inclusions - lighting which you dont find in the real world.
So in the end - try to find a stone within budget, my rules of thumb are H+ VS2+ and try to pick the one that sparkles the most (if the store stocks GIA graded stones, the try to pick/look at stones graded 'Excellent' or 'Very Good' for cut) and try to view the stone in different lighting conditions, away from the jewellery store lighting as that can make a lump of coal sparkle!
