Read performance when indexing one column vs indexing multiple columns (MySQL, InnoDB)

I have a table that has columns A, B and C. That table will have a lot of rows, let's say a million rows. I have a query that will be executed very often and will search the table by the A column, so I added an index on the A column.

I have another query that will not be executed as often as the first one but still often enough to qualify for a speed increase. That query will search by all the three rows.

My question is, will the second query have any speed increase from the index on column A or would I have to add indexes on all three columns in order to increase read speed? In other words, in a query where I have three WHERE conditions, will MySQL first search by the indexed column? Because if it does, out of those million rows, the first column would filter out maybe 100 or so, and querying those by the other two columns would be somewhat trivial and wouldn't require an index.
The index is a guide to the main table.  So your first query will use the Index on column A to identify which rows in the table have the data it wants, then does a key lookup to get the rest of the data and do further filtering if necessary.  

So your second query will also use Index A and then do a key lookup and then do further filtering as required.

But you can have just a single index on A,B and C and your first query will still use that index (providing that A is the first column indexed). 

So you can have your cake and eat it too.  I wouldn't create both indexes, I would just create the one on A,B and C.  However, If your table is narrow and only has four or five columns then I would only index on A.