Can I upgrade cassette without replacing rear derailleur? On my road bike, I have an old 8 speed Shimano 600 Cassette like this:


I'm not 100% sure about tooth count on each cog, but I think it's something like:
23, 21, 19,17,15, 14, 13,12 (probably?)

I'm having issues with the size of the cogs. Climbing even basic hills is very hard even on the biggest cog. (It's wearing out my knees). 

Can I change the cassette to one with bigger cogs without replacing my rear derailleur? Or would I have to get a new derailleur also?

While at it, can I get smaller front chain rings without replacing the front derailleur?
I have 53T/42T chain rings.
The derailleurs are usually matched to the size of the cassette/chainrings.  

the cage length of the rear derailleur is matched to the largest cog.  that is why you see mountain bikes with a 34 tooth in back with long cage derailleurs and road racing bikes with the little derailleurs matched to the corn cob size cassettes.

The curvature of the front derailleur is often matched to the size of the chain-rings.  This feature improves shifting. 

if the differences in tooth count are not too drastic, you might be able to get away with only swapping out cassette and/or chain rings.  Also the number of links in your chain may have to be adjusted.