Try to cut just deep enough to slice through the slingshot band without scratching the metal arm underneath it. A pocket knife with a serrated blade works well for this, but you can use any sharp blade you have available. For example, an X-ACTO knife or a boxcutter would also work.
You don’t have to be gentle or try to preserve the band in any way. Just pull and tear the band off the arm.
If you haven’t bought a replacement band yet, you can keep the old band to use as a reference when you’re shopping for a new one, if you want the same style of band.
Slingshot bands come in standard sizes that fit most slingshots. A normal band will cost you about $5 USD, while something like a high-velocity band can cost around $10.
Isopropyl alcohol is the recommended lubricant for this process because it is slippery enought to let you slide the band onto the arms, but will evaporate and dry up quickly, leaving the band securely attached to the arms. The higher the percentage of the alcohol, the faster it will dry after you put your new band on your slingshot. Use at least 70%-strength isopropyl alcohol. Use 90-99%-strength alcohol for the best results, if you can find it.
You can also splash some rubbing alcohol in your hand and rub it onto the slingshot arms.
Work fast after you lubricate the slingshot arm, so that the alcohol doesn’t drip off or dry up before you can get the band on.
It’s important to put the band all the way on the arms, past the bend in the arms, so that it will stay on securely when you use the slingshot. Don’t ever just slide the band onto the tips of the arms.