How can I find the sum of multiple groups within a series?
You can use the rolling method on a serie :
serie.rolling(24).sum()
To get directly the max 
max_idx = serie.rolling(24).sum().idxmax()
You range of interest is [max_idx-24+1:max_idx] (from index max_idx - 24 + 1 to index max_idx (both included) , so be careful if you want to retrieve these elements, with .loc should be fine but if going back to the original list or if using iloc then you'll have to go to max_idx + 1.
I have a short series of ~60 values. What I need to do is find the largest sum of 24 consecutive values in the series. 
e.g. I would need to be able to find the sums of the groups [0:23],[1:24],[2:25],[3:26], ... , [37:60] and determine which group has the largest sum.