I have a query with a number of test fields something like this:
SELECT * FROM some-table
WHERE field1 ILIKE \"%thing%\"
OR field2 ILIKE \"%thing\"
A full text search setup is not identical to a "contains" like query. It stems words etc so you can match "cars" against "car".
If you really want a fast ILIKE then no standard database index or FTS will help. Fortunately, the pg_trgm module can do that.