This is really an I/O subsystem issue, and less a query tuning issue. The load from your server exceeds what the I/O system can handle.

http://mssqlwiki.com/2012/08/27/io-requests-taking-longer-than-15-seconds-to-complete-on-file/

Tuning queries to use less reads can help, and indexing better might help, but this is a bit of an art. Really you need more/faster disks in the short term.

In looking at queries, I'd start with a workload trace, then feed that to the DTA and get an idea of what indexes it might recommend.

You can also take your trace and look for those queries with lots of reads in them (or writes) and see if there is something you can do to reduce the reads. Indexing helps here, but potentially rewriting SQL to work on smaller sets of data can help.