How to speed up SELECT .. LIKE queries in MySQL on multiple columns?

An index wouldn't speed up the query, because for textual columns indexes work by indexing N characters starting from left. When you do LIKE '%text%' it can't use the index because there can be a variable number of characters before text.

What you should be doing is not use a query like that at all. Instead you should use something like FTS (Full Text Search) that MySQL supports for MyISAM tables. It's also pretty easy to make such indexing system yourself for non-MyISAM tables, you just need a separate index table where you store words and their relevant IDs in the actual table.

Update

Full text search available for InnoDB tables with MySQL 5.6+.

An index won't help text matching with a leading wildcard, an index can be used for:

LIKE 'text%'

But I'm guessing that won't cut it. For this type of query you really should be looking at a full text search provider if you want to scale the amount of records you can search across. My preferred provider is Sphinx, very full featured/fast etc. Lucene might also be worth a look. A fulltext index on a MyISAM table will also work, but ultimately pursuing MyISAM for any database that has a significant amount of writes isn't a good idea.

An index can not be used to speed up queries where the search criteria starts with a wildcard:

LIKE '%text%'

An index can (and might be, depending on selectivity) used for search terms of the form:

LIKE 'text%'