Get last full backup and transaction log backup for each database
Something like this should get you the output that you're looking for.  The below query pulls the most recent backup of type full, differential, or log backup for each database.

;with backup_cte as
(
    select
        database_name,
        backup_type =
            case type
                when 'D' then 'database'
                when 'L' then 'log'
                when 'I' then 'differential'
                else 'other'
            end,
        backup_finish_date,
        rownum = 
            row_number() over
            (
                partition by database_name, type 
                order by backup_finish_date desc
            )
    from msdb.dbo.backupset
)
select
    database_name,
    backup_type,
    backup_finish_date
from backup_cte
where rownum = 1
order by database_name;