As you've seen by the answers, most folks do backup to disk first, then to tape. What I've often seen (and recommended) is to backup immediately to disk that is on the server itself (this could be SAN attached storage, the key is the backup isn't written across the network). Once this immediate backup is done, it is then copied to a central backup server. You keep an immediate copy of what you need to restore locally. On that backup server you keep several days worth of backups. That way if you do have to rollback to a previous day or two, you're not requesting a tape. And of course, you back up that central backup server to tape. So that covers your ability to recover.
With respect to the types of backup you should be doing, if you want to recover all transactions and be able to do what we call point in time recovery, you need to make sure the databases have their recovery model set to full recovery and you're going to want to do Full backups along with transaction log backups. You may want to intersperse differential backups to reduce the amount of files you are restoring. All of this is covered in books on-line and it'll give a better description than any of us can in a forum post.
As far as 3rd party products like LiteSpeed and Red Gate's SQL Backup, they used to be faster than native backup. That's because they use an API that SQL Server didn't. This was true in SQL Server 2000, but I'm not sure if it is still the case. However, they do encryption and compression and so may be well worth consideration given the size of your DBs.