SQL Indexing - ***BASIC****

Hoping someone could me with an ongoing indexing question that I have.

On my site, we have over the past 5 years developed what is emerging as a fairly complicated dbase structure, as features have been added to my site and relations have increased between different database tables, there has been a need to index fields in different ways, and in some instances field indexing has overlapped. For example we may have a table that has 5 fields (field1,field2,field3,field4,field5). A need to index field1 is requried because of a query that reads:

SELECT * From Table1 where field1=XXXXX

Additionally there may be a need to for another query that reads:

SELECT * From Table1 where field2=XXXXX

In this instance an index is placed on field2....
But, for example when there is the following query:

SELECT * From Table1 where field1=XXXXX and field2 = XXXXX

Is it necessary to set a new index on: field1,field2 ???

We have made the choice that yes, in fact there is...but now over time some of our tables have instances of single fields being indexed along with combinations of two single fields that have already been indexed, being indexed together. As tables have grown to over 1,000,000 records and having up to 15 or so indexes, we realize that the number of indexes maybe degrading performance. Also, indexes vary in type, e.g INT,BIGINT,Varchar fields... In the above instance, can we eliminate the multi-indexes and improve performance over all...?


On a second related question:

In the event that two tables are joined on a common field.

e.g. Select * from Table1,Table2 where Table1.field1=Table2.field1

Is it necessary to index both of these fields in tables: Table1 and Table2 ?


Hope someone can help, as we are looking to improve the efficiency of our tables as they continue to grow.