i have the original US kindle fire and the easiest way i've found to sideload apps was to either download the apk files from the net (just google for "appname apk" to find them) and then just email the apk file to myself and then on the kindle fire open my email and download the attached apk and open it. this way you don't need to mess around with trying to get a fiel maager on there in the first place as the files *should* just appear in your downloads list and the KF should allow you to open them from there.
unless amazon have somehow changed the way that works now?
another similar way is if you have another android based device (i.e. a phone or another tablet) to download an apk manager from the play store and install it, then you can use that to directly email any app on that android device to your email account for downloading and running as above.
fingers crossed it still works that way for you and you don't get any more headaches with it.
if it doesn't work, then the next stop will probably be to use ABD (google it) to get some kind of a file manager installed and you should be good to go from there via the emailing/downloading method.
thankfully the original kindle fire now has plenty of proper android roms out for it now, so i'm currently rocking a nice android jellybean rom which afaik was ported over from the nexus 7.