Production (20GB+) Database running slowly and timing out
First you need to figure out what the actual problem is. The sys.dm_os_wait_stats DMV can help with that. Most values in there are accumulative, so you need to capture it a few times  over an extended period to see what is actually going on. 

One of the things that this DMV can tell you is if you have RAM pressure. The graphic you posted is basically useless as SQL Server uses all memory it can get (if it needs to). So this is only showing that that mechanism is working.

Once you figured out what your biggest problem is, come back here to get more help. 



There are a few wait_types that can be considered noise. You can filter many of those out with this query:

SELECT wait_type,
       wait_time_ms / 1000.0 AS wait_in_sec,
       (wait_time_ms - signal_wait_time_ms) / 1000.0 AS resource_wait_sec,
       signal_wait_time_ms / 1000.0 AS signal_sec,
       waiting_tasks_count AS wait_count,
       100.0 * wait_time_ms / SUM (wait_time_ms) OVER() AS percentage
    FROM sys.dm_os_wait_stats
    WHERE wait_type NOT IN (
        'CLR_SEMAPHORE', 'LAZYWRITER_SLEEP', 'RESOURCE_QUEUE', 'SLEEP_TASK',
        'SLEEP_SYSTEMTASK', 'SQLTRACE_BUFFER_FLUSH', 'WAITFOR', 'LOGMGR_QUEUE',
        'CHECKPOINT_QUEUE', 'REQUEST_FOR_DEADLOCK_SEARCH', 'XE_TIMER_EVENT', 'BROKER_TO_FLUSH',
        'BROKER_TASK_STOP', 'CLR_MANUAL_EVENT', 'CLR_AUTO_EVENT', 'DISPATCHER_QUEUE_SEMAPHORE',
        'FT_IFTS_SCHEDULER_IDLE_WAIT', 'XE_DISPATCHER_WAIT', 'XE_DISPATCHER_JOIN', 'BROKER_EVENTHANDLER',
        'TRACEWRITE', 'FT_IFTSHC_MUTEX', 'SQLTRACE_INCREMENTAL_FLUSH_SLEEP')
     AND waiting_tasks_count &gt;0
     AND wait_time_ms &gt;0
  ORDER BY wait_time_ms DESC;


(based on http://www.sqlskills.com/blogs/paul/survey-what-is-the-highest-wait-on-your-system/)