Depending on the coral you may or may not need any tools at all.
SPS are probably the easiest ones to frag, you just break a branch of. This can be done with fingers if you can grab it. I use bone cutters, but not for their cutting power, but to simply grasp the branch and then turn the cutters. This usually creates a break right below when cutters are holding onto the coral. Which means I'm still holding on to the frag and can pull it out from the tank. If I were to actually cut with them, frag would be free falling after the break if not launched with some force and can be lost if you are doing this in the tank and it falls into a hard to reach spot
For stuff like pally/zoa, I try to break the rock they are sitting on. I can do this in most cases with the same bone cutters. This way corals are still attached to something. Attaching just the zoa part to a rock/frag plug can be a bit tricky as they will tend to pull away from the glue. If you do end up with loose zoa/pally, throw some small rocks into a container and float coral on top, it will take maybe a week or two, but eventually it will attach on its own. The container is of course left in the tank for the period and is mainly used to make sure that the frag stays put next to the rocks you want it to attach to.
Mushrooms - are cut in pieces with cuts going through the center, if you are trying to actually frag individual piece, otherwise same as zoa pally, try to break a rock coral is attached to, if not use container method to attach later.
Favia/acan type corals, here you will need to cut them with a saw most likely. Although if your colony is big enough you could try breaking it somehow. I haven't done this in a while, but I fragged these with a dremel. If long straight cut is enough you could use a hand saw of some sort, or even one of those saws with thin blade, then you could cut in any path you want. Either type of coral is pretty durable so trying to cut around mouth is not really needed, they will likely heal up and form two heads out of one that was cut.
Encrusting things like monties, chalices etc can usually be pried off in small chunks as they do not attach to the rock too well, but that is matter of luck. So could use some saw cutting, which again can be done with simple tools. For chalice frags try to get few eyes/mouths in there.
In the end there really isn't a need for lot of tools to start fragging with. A small pair of pliers would be enough for SPS and a saw with thin blade for other things. Knife/razor could be helpful for the softies.
Finally do not be afraid to frag. As long as your tank is stable and your corals are healthy if it very easy and safe for the corals in most cases.
If you can say which coral types you are actually fragging perhaps we can give a more specific advice on that.