I have a question about backups. If you setup transaction log backups only is sql management studio smart enough to take a full backup when it needs it to have the transaction logs afterwards? Right now I have transaction log backups running every two hours with an expiration ten days. I also have avamar backup running a full sql backup nightly. But can I do a transaction log restore through sql management studio without management studio ever set to do its own full backups?
I don't know what Avamar backup is - seems it is a disk replication/backup. For you to restore a database, you need a full backup and all the log files since the full backup. So the answer most likely is that you cannot restore without also having a full database backup. SQL Server does not automatically schedule or take full backups simply because you set up log backups.
So what I should probably do is have SQL server take a full backup every evening and then the transaction log backups can work throughout the day until the next full backup that following evening?
Yes. The frequency of the full backups and the log backups depends on your environment, but in general that is what is required.
Given that backing up the database and restoring and recovering when required is a major function of the DBA, you might want to do some research/education into the backup/restore process and best practices. Here are some useful links:
http://www.brentozar.com/sql/backup-best-practices/
http://minionware.net/backup/
https://ola.hallengren.com/