How is my database being backed up?
In general you should have full backups set on one schedule, and transaction log backups set on another (more frequent) schedule, so yes do both of these as soon as you can.

The issue you will run into is if another process is performing backups without "COPYONLY", as this will through off sequencing (making a full point in time restore impossible unless you can track down the location of the mystery backups).

The code snippet below will search the default trace for all BACKUP activity and should allow you to track things down.

SET NOCOUNT ON;
--this version reads all trace files
declare @path  nvarchar(100)
set @path = (select top 1 [path] from sys.traces where is_default = 1)
set @path = (select reverse(right(reverse(@path), (Len(@path) - (PATIndex('%[_]%', reverse(@path)))))) + '.trc')


SELECT  STE.name AS EventClassName,
       ST.StartTime ,
        ST.LoginName ,
        ST.HostName ,
        ST.ApplicationName ,
        ST.TextData
FROM    sys.fn_trace_gettable((@path), DEFAULT) ST
INNER JOIN sys.trace_events STE ON ST.EventClass = STE.trace_event_id
WHERE   TextData LIKE '%backup%' AND SPID &lt;&gt; @@SPID
ORDER BY StartTime DESC;