You really only need two tools to remove a horseshoe: An old rasp and an old/cheap pair of nippers.
Any trimmer or farrier could supply you with the old rasp (I've got one you could have) and they might have an old pair of nippers they'll give you, otherwise just buy a cheap pair. Cheap nippers are cheap because they don't cut particularly well but that's okay, you don't need them to cut to remove a shoe.
Get the horse on a hard, solid surface like a concrete floor, rubber mat, pavement, or a piece of plywood; plop some hay in front of their face (this will take a while); and grab that rasp. While the horse stands on the foot, use the narrow edge of the rasp to carefully saw off the nail clinches. Hold the rasp parallel to the hoofwall and saw downwards on the clinches one-at-a-time until they are just shiny little squares flush with the hoofwall. Try to take off as little hoofwall as possible. It's very easy to dig a ditch in the hoofwall while you're rasping those clinches so check often to make sure you aren't doing this- if you are then you need to change the angle of your rasp.
Make sure not to position yourself directly in front of your horse's leg. It's very easy to get clocked in the head by a knee if you don't watch what you're doing.
Once all your clinches are off, pick up the leg and carefully place the teeth of the nippers between the shoe and the hoofwall at one heel behind the last nail. Don't cut into the hoofwall! Once you've got the nippers positioned pull the handle down and inwards toward the toe. When you've got it loosened a little switch sides. Keep switching sides until those last nails start popping out of the hoof, then position your nippers in front of those nails and continue on until all the nails have loosened up enough for the shoe to come off.
Voila, you've removed a horseshoe!
What is nice about this method is that you don't have to do any banging on the hoof to get the shoe off. If the horse is laminitic or suffering from an injury, reducing the amount of tools pounding on the foot really makes a difference.
This first video shows a slightly different method. Notice that she just rasps the whole hoofwall to remove the clinches. This works but it kills your rasp and takes off a lot of excess hoofwall. I would also NOT recommend using a claw hammer like she shows unless you absolutely have to. It would be too easy to not get it anchored well, pull too hard, and end up with a claw hammer in your face or your horse. EEK!
This second video is hawking a product, an emergency shoe pulling kit, and shows how to use a tool to pull the nails out one by one. That's fine if your horse doesn't mind the banging, but it's not necessary. I put it up because I enjoyed watching this woman with her nicely manicured nails and shiny tall boots pull a shoe without even using gloves to hold the rasp. She's more hardcore than she looks!
Hopefully you'll be lucky and never have to do this, but in case you do- it doesn't hurt to keep a rasp and nippers on hand!