Sounds like you don't understand how backups in SQL Server work. There are three different kinds of backups in SQL Server.
FULL - Full database backups dump the entire database and the active transaction log to a single (or striped, but lets ignore this feature for now) file on disk.
DIFFERENTIAL - Differential database backups backup all the data which has been changed since the last full backup to a backup file. To restore from a differential backup you MUST have the full backup which was taken before the differential backup was taken.
LOG - Log backups backup just the data which is currently in the transaction log. Log backups give you the ability to restore the database to a specific point it time (for example if someone dropped a table you could restore to a point just restore the table was dropped and get back all the data from that table). LOG backups require that your database be in either FULL or BULK_LOGGED recovery mode. To restore log backups you have to have the most recent full backup taken before the log backup you want to restore, optionally the differential backup taken between the full backup and the log backup, and all the log backups between either the full or differential backup and the last log file you want to restore to.
From what you have described it sounds like you need to loose as little data as possible. You'll need to change the recovery mode to FULL or BULK_LOGGED (FULL is what most people select) and then setup daily full backups and transaction log backups every few minutes. The frequency that you run the log backups will depend on how much data you can afford to loose. The business should be the one that sets this number, not you. It isn't uncommon to run log backups every 10-15 minutes depending on the requirements.
After the backups are setup, you'll want to configure something to delete the files after some number of days (depending on the business requirements and how much space you can afford to store).
You'll want to backup the databases to a server in another EC2 availability zone so that if a repeat of the Amazon outage from earlier this year happens your backup data is still safe.
It might be worth it to bring a consultant in for an hour or two to make sure that your backups are done correctly. If they aren't done right, now is the time to figure that out, not when the system has failed.