You can search your notes in GoldMine using the Contact Search Center.
Open the Contact Search Center (Lookup >> Contact).
For the 'Lookup' field, from the drop down list, select 'Notes'.
Similarly, for the 'that' field, select 'Contains' from the drop down list.
In the last field, type in the word that you are searching for and hit 'Find'.
The result will list all contacts whose notes contain the word you have searched for.
Note: To search for multiple words in notes, click on the 'Insert Condition' button, choose And/Or and input the next word you want to search for.