If you've been using the Mac for several days -after- the unwanted file deletions, there is a chance (just a chance, but in fact a possibility) that the "deleted" files (the sectors of the drive that contained the files) have already been "over-written" by -new- data. In that case they are unrecoverable and GONE.
There aren't any "easily-usable" file recovery apps out there that are free. The ones that are easy to use all cost money.
Here are names to investigate:
- DataRescue3
- Stellar Phoenix Data Recovery
- Disk Drill
- Nice to Recover
ALL of these can be freely downloaded and will run in "demo mode". Demo mode means that the recovery app will scan the drive, and build a list of all the "recoverable" files that it finds. It will also let you recover -one- file, just to see if recovery is in fact possible. If this works, you then pay the registration fee, get a code, enter the code, and the app will "go to work" on the drive.
Again, the chances of successful recovery are significantly diminished because you have continued to use the drive after the deletions.
If you don't know how to properly back up, it's time to learn. Many here are willing to teach.
My suggestion:
Get a cheap "USB/SATA" dock (can be found for $20-25 at amazon) and a "bare" SATA drive to go into it (someone may even have an old drive they will give you for free). Then use the -free- "CarbonCopyCloner" app to "clone" the contents of your main drive to the docked drive.
Do this every 3 days or so (only takes a few mins), and you will be pretty-much protected against drive disasters. The cloned drive is an exact copy of your internal drive and is also BOOTABLE -- very important in a moment of extreme need when you can't get your Mac up and running. (Time Machine backups are NOT bootable, and the files on a TM backup are NOT in POFF [plain ol' finder format])
CCC is the better way to backup.