Sort by date field

I have developers asking why is the query select * from table_name order by date; running so slow. I looked at the table. The table has three indexes, but date field is not in any of the three indexes. My question is if the date field is not indexes will the query be doing full table scan even though there are indexes on id field? Another question is, will it be a good idea to index the date field? Please advise. Thank you very much.