Need a Walkthrough on How to Create Database & Transaction Log Backups

Question

Is this the proper forum to ask for this type of guidance?  There has been bad blood between my department (Research) and the MIS department for 30 years, and long story short I have been "given" a virtual server and cut loose by my MIS department
-- it's my responsibility for installs, updates, backups, etc.  I have everything running really well, I believe, with the exception of my transaction log backups -- my storage unit is running out of space on a daily basis, so I feel like I have to be
doing something wrong.
If this is the proper forum, I'll supply the details of how I currently have things set up, and I'm hoping with some loving guidance I can work the kinks out of my backup plan.  High level -- this is for a SQL Server 2012 instance running on a Windows
2012 Server...

Answer 1

Thanks all, after posting this I'm going to read the materials provided above.  As for the details:
I'm running on a virtual Windows Server 2012 Standard, Intel Xeon CPU 2.6 GHz with 16 GB of RAM; 64 bit OS.  The computer name is e275rd8
Drives (NTFS, Compression off, Indexing on):
DB_HVSQL_SQL-DAT_RD8-2(E:) 199 GB (47.2 used; 152 free)
DB_HVSQL_SQL-Dat_RD8(F:) 199 GB (10.1 used; 189 free)
DB_HVSQL_SQL-LOG_RD8-2(L:) 199 GB (137 used; 62 free) **
DB_HVSQL_SQL-BAK_RDu-2(S:) 99.8 GB (64.7 used; 35 free)
DB_HVSQL_SQL-TMP_RD8-2(T:) 99.8 GB (10.6 used; 89.1 free)
SQL Server:
Product: SQL Server Enterprise (64-bit)
OS: Windows NT 6.2 (9200)
Platform: NT x64
Version: 11.0.5058.0
Memory: 16384 (MB)
Processors: 4
Root Directory: f:\Program Files\Microsoft SQL Server\MSSQL11.MSSQLSERVER\MSSQL
Is Clustered: False
Is HADR Enabled: False
Database Settings:
Default index fill factor: 0
Default backup media retention (in days): 0
Compress backup is checkmarked/on
Database default locations:
  Data: E:\SQL\Data
  Log: L:\SQL\LOGs
  Backup: S:\SQLBackups
There is currently only one database: DistrictAssessmentDW
To create my backups, I'm using two maintenance plans, and this is where I'm pretty sure I'm not doing something correctly.  My entire setup is me just guessing what to do, so feel free to offer suggestions...
Maintenance Plan #1: Backup DistrictAssessmentDW
  Scheduled to run daily Monday Through Friday at 3:33 AM
  Step 1: Backup Database (Full) 
    Backup set expires after 8 days 
    Back up to Disk (S:\SQLBackups)
    Set backup compression: using the default server setting
  Step 2: Maintenance Cleanup Task
    Delete files of the following type: Backup files
    Search folder and delete files based on an extension:
      Folder: L:\SQL\Logs
      File extension: trn
      Include first-level subfolders: checkmarked/on
    File age: Delete files based on the age of the file at task run time older than 1 Day
  Step 3: Maintenance Cleanup Task
    Delete files of the following type: Backup files
    Search folder and delete files based on an extension:
      Folder: S:\SQLBackups
      File extension: bak
      Include first-level subfolders: checkmarked/on
    File age: Delete files based on the age of the file at task run time older than 8 Days
Maintenance Plan #2: Backup DistrictAssessmentDW TRANS LOG ONLY
  Scheduled to run daily Monday through Friday; every 20 minutes starting at 6:30 AM & ending at 7:00 PM
  Step 1: Backup Database Task
    Backup Type: Transaction Log
    Database(s): Specific databases (DistrictAssessmentDW)
    Backup Set will expire after 1 day
    Backup to Disk (L:\SQL\Logs\)
    Set backup compression: Use the default server setting
Around 2:30 each day my transaction log backup drive (L:) runs out of space.  As you can see, transactions are getting backed up every 20 minutes, and the average size of the backup files is about 5,700,000 KB.
I hope this covers everything, if not please let me know what other information I need to provide...