Correcting a database that has had indexes and statistics created on it willy-nilly I've inherited a DB and I believe it was the victim of DBAs who encountered random slow operations, ran the tuning adviser for individual queries and blindly created the recommend indexes and statistics for said individual cases with no concern for the impact on the DB overall.
How do I remedy this situation? 

I myself am not a DBA but am interested in doing this properly - 


Should I drop all indexes and statistics and start over? 
Is there a tool out there that can look at the entirety of SQL operations and recommend which indexes and stats to delete, update or create. 
Any recommended standard procedures for this situation, I can't imagine I am unique in encountering this scenario


I'm on SQL Server 2012
Consider following the STOP pattern that's used in first aid training. Stop, think, observe and plan. That is, don't rush into action. Start by gathering some information, draft an action plan and only then start doing things. If there's an electricity accident, one doesn't touch the victim before cutting power. Likewise, one shouldn't start dropping indexes without knowing what use those would be.

Is the system working well enough? How can you tell? That is, what's the definition of "well enough?"

Do you have a set of business rules like "Report X must be ready within Y minutes" and "Daily ETL must be finished by 0600 as cube update starts at 0610." Maybe there's something like "Database backup must be finished at 0200, as disk-to-tape backup starts at 0300." Be aware there might be some ad-hoc rules like "CFO needs $stuff NOW" that are usually not documented at all.

If you don't have any business constraints, you need to discuss with your boss, other users and business about such constraints.

If you've got a set of rules, that's great news. Now, see if the system respects those limits at the moment.

After you know the rules for the system, maybe it would be a great idea to do some basic health check. Brent Ozar's sp_Blitz is widely used free tool. There's warnings if backups are not handled properly, lack of recent integrity checks, configuration issues and a lot more.