You can choose the option to allow TheSage to run automatically at startup. If so, it's always immediately available for the type of searches you refer to
Then, to look up a word from a browser's display (or from any other program) you can place the cursor over a word and then press the key combination that calls up TheSage's definitions. (By default that key combination is Ctrl+Shift+A but you can change it in the options if you want to). I actually find that highlighting the word first (by double-clicking on it) is more reliable than simply placing the cursor over it.
(Theoretically you can also use a similar mouse-based technique to the one you've been used to but it doesn't seem to work for me)
Once you've done that you'll see all of the main definitions for the word listed in a tree structure. Click on 'T' alongside a definition provides you with synonym, antonyms, hyponyms and hypernyms for that definition. Clicking on 'E' alongside a definition provides you with examples of the word used in phrases and sentences.
There's far more to it than that but, as with many programs, you just need to select the tools that are useful to you.
Chris