What groupsets are compatible with my bike?
There are a lot of things to check when doing a swap of your groupset.  Some things off the top of my head:


Hub/Freewheel compatibility:  Since you are increasing the cassette size, you may have to get a different hub and/or freewheel.  If the freewheel needs to be changed out, the new one may end up being a wider dimension, which will require a new hub that will keep the overall dimension of the hub/freewheel the same.  On mountain bikes, you may also have to contend with different axle designs as well.  i.e. through-axle vs. quick-release and different dimensions between chain stays.  
Alignment of the chain-rings with the cassette: If you are changing the number of chain-rings or changing the spacing significantly, you will need to check that the spacing between the frame and rings is adjusted with shims on the bottom bracket or a new bottom bracket design entirely.  If you have braze-on derailleur hangers, this is critical.  If they are clamp derailleur hangers, there is some adjustment available.
Bottom bracket:  Most groups sets include the bottom bracket.  You will need to check that the frame diameter and fitment style (press-in vs. thread-in) will match the new bottom bracket
Shifters:  Since brake levers and shifters are typically integrated on a road bike, you will need to replace them to accommodate the new indexing on the rear cassette and/or front chain-rings.  Make sure that the new brake levers are compatible with your existing brake calipers.  Should be pretty standard on road bikes, but mountain bikes and hybrids require some more investigation when using integrated shifters.  


Overall, you have a much better chance of getting a compatible upgrade if you stick with the same brand and similar technology (cable operated brakes vs. hydraulic for example)

Note:  As OP suggested, this answer was as generic as I could make it to fit other bike types/styles.